Download LECTURE NO.19 Gauss`s law

LECTURE NO: 01 INTRODUCTION
In this set of approximately 49 lectures covering a
course of one semester, I would take you through electrostatics,
magneto statics and electromagnetic phenomena, leading to
both the differential and integral form of the Maxwell’s
equations. At the end of the course you would have an
appreciation of what are the important phenomena and
problems associated with electromagnetism.
However, the course requires a good understanding of
the subject of vector calculus. So in the initial lectures, we
would spend some time in revising or providing an
introduction to the essentials of vector calculus. It is not going
to be rigorous the way a mathematician would like it to be but
should adequately serve our purpose. In the first module, we
will discuss vector calculus and some of its basic applications.
We will have discussions on the concept of a scalar field and a
vector field, ordinary derivatives and gradient of a scalar
function, line and surface integrals, divergence and curl of a
vector field, Laplacian. We will enunciate two major theorems,
viz., the divergence theorem and the Stoke’s theorem.
Concept of a Field:
By field, we basically mean something that is
associated with a region of space. For instance, this room in
which I am speaking can be considered to be a region in which
a temperature field exists. Normally, we talk of the temperature
of a room. However, this is in the sense of an average and does
not provide detailed temperature profile inside the room.
However, the temperature inside a room does vary from place
to place. For instance, if you are in a kitchen, the temperature
would be higher when you are close to stove and would be
lower elsewhere. In principle, one can associate a temperature
with every point inside the room. The field that we talked of
here, viz. the temperature field is a scalar field because the
field quantity “temperature” is a scalar.
The “field” is thus a region of space where with every
point we can associate a scalar or a vector (it could be more
generalized but for our purposes, these two will do). Coming to
a vector field, as we know, a vector quantity has both
magnitude and direction. Consider our room again. We can
associate a gravitational field with it. Though we generally say
that the acceleration due to gravity has a constant value inside
the room, it is also meant in an average sense. In reality, its
value and direction differs from place to place and a mass
inside a room experiences a different force (both in magnitude
and direction) depending on where in the room it is placed. If
we talk of associating a force with every point in a certain
region of space, we are talking about a vector field. In 2
dimensions, the force is a function of positions x and y and in
three dimensions it is a function of x, y and z. Other than
gravitational field, examples of vector fields are electric field
and magnetic field.
Text Book:
1. Matthew
N.
O.
th
Sadiku,
Principles
of
Electromagnetics, 4 Ed., Oxford Intl. Student Edition.
Reference Book:
1. C. R. Paul, K. W. Whites, S. A. Nasor, Introduction to
Fig : Illustration of a Cartesian coordinate plane.
rd
Electromagnetic Fields, 3 ,TMH.
th
2. Electromagnetic Field Theory, W.H. Hyat, TMH, 7
Ed.
3. Engineering Electromagnetics by Shen, Kong,
Patnaik, CENGAGE Learning.
LECTURE NO: 02 CARTESIAN COORDINATE
SYSTEM
A Cartesian coordinate system is a coordinate system
that specifies each point uniquely in a plane by a pair of
numerical coordinates, which are the signed distances from the
point to two fixed perpendicular directed lines, measured in the
same unit of length. Each reference line is called a coordinate
axis or just axis of the system, and the point where they meet is
its origin, usually at ordered pair (0, 0). The coordinates can
also be defined as the positions of the perpendicular projections
of the point onto the two axes, expressed as signed distances
from the origin
Four points are marked and labeled with their coordinates:
(2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin
(0,0) in purple.
One can use the same principle to specify the position
of any point in three-dimensional space by three Cartesian
coordinates, its signed distances to three mutually
perpendicular planes (or, equivalently, by its perpendicular
projection onto three mutually perpendicular lines). In general,
n Cartesian coordinates (an element of real n-space) specify the
point in an n-dimensional Euclidean space for any dimension n.
These coordinates are equal, up to sign, to distances from the
point to n mutually perpendicular hyperplanes.
Cartesian coordinate system with a circle of radius 2 centered
at the origin marked in red. The equation of a circle is (x − a)2
+ (y − b)2 = r2 where a and b are the coordinates of the center
(a, b) and r is the radius.
One dimension
Choosing a Cartesian coordinate system for a one-dimensional
space that is, for a straight line involves choosing a point O of
the line (the origin), a unit of length, and an orientation for the
line. An orientation chooses which of the two half-lines
determined by O is the positive, and which is negative; we then
say that the line "is oriented" (or "points") from the negative
half towards the positive half. Then each point P of the line can
be specified by its distance from O, taken with a + or − sign
depending on which half-line contains P.
A line with a chosen Cartesian system is called a number line.
Every real number has a unique location on the line.
Conversely, every point on the line can be interpreted as a
number in an ordered continuum such as the real numbers.
The choices of letters come from the original convention,
which is to use the latter part of the alphabet to indicate
unknown values. The first part of the alphabet was used to
designate known values.
In the Cartesian plane, reference is sometimes made to a unit
circle or a unit hyperbola.
Three dimensions
Two dimensions
The modern Cartesian coordinate system in two dimensions
(also called a rectangular coordinate system) is defined by an
ordered pair of perpendicular lines (axes), a single unit of
length for both axes, and an orientation for each axis. (Early
systems allowed "oblique" axes, that is, axes that did not meet
at right angles.) The lines are commonly referred to as the xand y-axes where the x-axis is taken to be horizontal and the yaxis is taken to be vertical. The point where the axes meet is
taken as the origin for both, thus turning each axis into a
number line. For a given point P, a line is drawn through P
perpendicular to the x-axis to meet it at X and second line is
drawn through P perpendicular to the y-axis to meet it at Y. The
coordinates of P are then X and Y interpreted as numbers x and
y on the correspondingnumber lines. The coordinates are
written as an ordered pair (x, y).
The point where the axes meet is the common origin of the two
number lines and is simply called the origin. It is often labeled
O and if so then the axes are called Ox and Oy. A plane with xand y-axes defined is often referred to as the Cartesian plane or
xy plane. The value of x is called the x-coordinate or abscissa
and the value of y is called the y-coordinate or ordinate.
A three dimensional Cartesian coordinate system, with origin
O and axis lines X, Y and Z, oriented as shown by the arrows.
The tick marks on the axes are one length unit apart. The black
dot shows the point with coordinates x = 2, y = 3, and z = 4, or
(2,3,4).
Choosing a Cartesian coordinate system for a threedimensional space means choosing an ordered triplet of lines
(axes) that are pair-wise perpendicular, have a single unit of
length for all three axes and have an orientation for each axis.
As in the two-dimensional case, each axis becomes a number
line. The coordinates of a point P are obtained by drawing a
line through P perpendicular to each coordinate axis, and
reading the points where these lines meet the axes as three
numbers of these number lines.
Alternatively, the coordinates of a point P can also be taken as
the (signed) distances from P to the three planes defined by the
three axes. If the axes are named x, y, and z, then the xcoordinate is the distance from the plane defined by the y and z
axes. The distance is to be taken with the + or − sign,
depending on which of the two half-spaces separated by that
plane contains P. The y and z coordinates can be obtained in
the same way from the x–z and x–y planes respectively.
Fig :The four quadrants of a Cartesian coordinate system.
The coordinate surfaces of the Cartesian coordinates (x, y, z).
The z-axis is vertical and the x-axis is highlighted in green.
Thus, the red plane shows the points with x = 1, the blue plane
shows the points with z = 1, and the yellow plane shows the
points with y = −1. The three surfaces intersect at the point P
(shown as a black sphere) with the Cartesian coordinates (1,
−1, 1).
Quadrants and octants
Main articles: Octant (solid geometry) and Quadrant (plane
geometry)
The axes of a two-dimensional Cartesian system divide the
plane into four infinite regions, called quadrants, each bounded
by two half-axes. These are often numbered from 1st to 4th and
denoted by Roman numerals: I (where the signs of the two
coordinates are +,+), II (−,+), III (−,−), and IV (+,−). When the
axes are drawn according to the mathematical custom, the
numbering goes counter-clockwise starting from the upper
right ("north-east") quadrant.
Similarly, a three-dimensional Cartesian system defines a
division of space into eight regions or octants, according to the
signs of the coordinates of the points. The convention used for
naming a specific octant is to list its signs, e.g. (+ + +) or (− +
−). The generalization of the quadrant and octant to arbitrary
number of dimensions is the orthant, and a similar naming
system applies.
Cartesian formulas for the plane
Distance between two points
The Euclidean distance between two points of the plane with
Cartesian coordinates
and
is
Rotation
To rotate a figure counterclockwise around the origin by some
angle is equivalent to replacing every point with coordinates
(x,y) by the point with coordinates (x',y'), where
This is the Cartesian version of Pythagoras's theorem. In threedimensional
space,
the
distance
between
points
and
is
Thus:
which can be obtained by two consecutive applications of
Pythagoras' theorem.
Thus:
General matrix form of the transformations
Euclidean transformations
These Euclidean transformations of the plane can all be
described in a uniform way by using matrices. The result
of applying a Euclidean transformation to a point
The Euclidean transformations or Euclidean motions
are the (bijective) mappings of points of the Euclidean plane to
themselves which preserve distances between points. There are
four types of these mappings (also called isometries):
translations, rotations, reflections and glide reflections.
Translation
Translating a set of points of the plane, preserving the distances
and directions between them, is equivalent to adding a fixed
pair of numbers (a, b) to the Cartesian coordinates of every
point in the set. That is, if the original coordinates of a point
are (x, y), after the translation they will be
is given by the formula
where A is a 2×2 orthogonal matrix and b = (b1, b2) is an
arbitrary ordered pair of numbers;[6] that is,
where
[Note the use of row vectors for point coordinates and that the
matrix is written on the right.]
To be orthogonal, the matrix A must have orthogonal rows
with same Euclidean length of one, that is,
A11A21+A12A22=0
And
A211+A212=A221+A222=1
This is equivalent to saying that A times its transpose must be
the identity matrix. If these conditions do not hold, the formula
describes a more general affine transformation of the plane
provided that the determinant of A is not zero.
The formula defines a translation if and only if A is the identity
matrix. The transformation is a rotation around some point if
and only if A is a rotation matrix, meaning that
A11A22- A21A12=1
A reflection or glide reflection is obtained when,
A11A22- A21A12=-1
Assuming that translation is not used transformations can be
combined by simply multiplying the associated transformation
matrices.
In two dimensions
Fig :The right hand rule.
Fixing or choosing the x-axis determines the y-axis up to
direction. Namely, the y-axis is necessarily the perpendicular to
the x-axis through the point marked 0 on the x-axis. But there
is a choice of which of the two half lines on the perpendicular
to designate as positive and which as negative. Each of these
two choices determines a different orientation (also called
handedness) of the Cartesian plane.
The usual way of orienting the axes, with the positive x-axis
pointing right and the positive y-axis pointing up (and the xaxis being the "first" and the y-axis the "second" axis) is
considered the positive or standard orientation, also called the
right-handed orientation.
A commonly used mnemonic for defining the positive
orientation is the right hand rule. Placing a somewhat closed
right hand on the plane with the thumb pointing up, the fingers
point from the x-axis to the y-axis, in a positively oriented
coordinate system.
The other way of orienting the axes is following the left hand
rule, placing the left hand on the plane with the thumb pointing
up.
Fig :3D Cartesian Coordinate Handedness
When pointing the thumb away from the origin along an axis
towards positive, the curvature of the fingers indicates a
positive rotation along that axis.
Switching any two axes will reverse the orientation, but
switching both will leave the orientation unchanged.
In three dimensions
Fig. 5: The left-handed orientation is shown on the left, and the
right-handed on the right.
Fig.: The right-handed Cartesian coordinate system indicating
the coordinate planes.
Once the x- and y-axes are specified, they determine the
line along which the z-axis should lie, but there are two
possible directions on this line. The two possible coordinate
systems which result are called 'right-handed' and 'left-handed'.
The standard orientation, where the xy-plane is horizontal and
the z-axis points up (and the x- and the y-axis form a positively
oriented two-dimensional coordinate system in the xy-plane if
observed from above the xy-plane) is called right-handed or
positive.
The name derives from the right-hand rule. If the index
finger of the right hand is pointed forward, the middle finger
bent inward at a right angle to it, and the thumb placed at a
right angle to both, the three fingers indicate the relative
directions of the x-, y-, and z-axes in a right-handed system.
The thumb indicates the x-axis, the index finger the y-axis and
the middle finger the z-axis. Conversely, if the same is done
with the left hand, a left-handed system results.
Representing a vector in the standard basis
A point in space in a Cartesian coordinate system may
also be represented by a position vector, which can be thought
of as an arrow pointing from the origin of the coordinate
system to the point. If the coordinates represent spatial
positions (displacements), it is common to represent the vector
from the origin to the point of interest as . In two dimensions,
the vector from the origin to the point with Cartesian
coordinates (x, y) can be written as:
r  xi  y j
1
 0
i    and j   
 0
1
where, iand j are unit vectors in the direction of the x-axis and
y-axis respectively, generally referred to as the standard basis
(in some application areas these may also be referred to as
versors). Similarly, in three dimensions, the vector from the
origin to the point with Cartesian coordinates (x,y,z)can be
written as:
 0
 
r  xi  y j  z k where k   0  is the unit vector in the
1
 
direction of the z-axis.
There is no natural interpretation of multiplying vectors to
obtain another vector that works in all dimensions, however
there is a way to use complex numbers to provide such a
multiplication. In a two dimensional cartesian plane, identify
the point with coordinates (x, y) with the complex number z =
x + iy. Here, i is the imaginary unit and is identified with the
point with coordinates (0, 1), so it is not the unit vector in the
direction of the x-axis. Since the complex numbers can be
multiplied giving another complex number, this identification
provides a means to "multiply" vectors. In a three dimensional
cartesian space a similar identification can be made with a
subset of the quaternions.
LECTURE NO.03 CYLINDRICAL COORDINATE
SYSTEM
A cylindrical coordinate system is a three-dimensional
coordinate system that specifies point positions by the distance
from a chosen reference axis, the direction from the axis
relative to a chosen reference direction, and the distance from a
chosen reference plane perpendicular to the axis. The latter
distance is given as a positive or negative number depending
on which side of the reference plane faces the point.
The axis is variously called the cylindrical or longitudinal axis,
to differentiate it from the polar axis, which is the ray that lies
in the reference plane, starting at the origin and pointing in the
reference direction.
The distance from the axis may be called the radial distance or
radius, while the angular coordinate is sometimes referred to as
the angular position or as the azimuth. The radius and the
azimuth are together called the polar coordinates, as they
correspond to a two-dimensional polar coordinate system in the
plane through the point, parallel to the reference plane. The
third coordinate may be called the height or altitude (if the
reference plane is considered horizontal), longitudinal position,
or axial position.
Cylindrical coordinates are useful in connection with objects
and phenomena that have some rotational symmetry about the
longitudinal axis, such as water flow in a straight pipe with
round cross-section, heat distribution in a metal cylinder,
electromagnetic fields produced by an electric current in a
long, straight wire, and so on.
Definition
The three coordinates (ρ, φ, z) of a point P are defined as:

A cylindrical coordinate system with origin O, polar axis A,
and longitudinal axis L. The dot is the point with radial
distance ρ = 4, angular coordinate φ = 130°, and height z = 4.
The origin of the system is the point where all three
coordinates can be given as zero. This is the intersection
between the reference plane and the axis.


The radial distance ρ is the Euclidean distance from the
z axis to the point P.
The azimuth φ is the angle between the reference
direction on the chosen plane and the line from the
origin to the projection of P on the plane.
The height z is the signed distance from the chosen
plane to the point P.
Unique cylindrical coordinates
As in polar coordinates, the same point with cylindrical
coordinates (ρ, φ, z) has infinitely many equivalent
coordinates, namely (ρ, φ ± n×360°, z) and (−ρ, φ ± (2n +
1)×180°, z), where n is any integer. Moreover, if the radius ρ is
zero, the azimuth is arbitrary.
plane shows the points with z=1, and the yellow half-plane
shows the points with φ=−60°. The z-axis is vertical and the xaxis is highlighted in green. The three surfaces intersect at the
point P with those coordinates (shown as a black sphere); the
Cartesian coordinates of P are roughly (1.0, −1.732, 1.0).
In situations where one needs a unique set of coordinates for
each point, one may restrict the radius to be non-negative
(ρ ≥ 0) and the azimuth φ to lie in a specific interval spanning
360°, such as (−180°,+180°] or [0,360°].
Cylindrical Coordinate Surfaces. The three orthogonal
components, ρ (green), φ (red), and z (blue), each increasing at
a constant rate. The point is at the intersection between the
three colored surfaces.
Conventions
The notation for cylindrical coordinates is not uniform
(ρ, φ, z), where ρ is the radial coordinate, φ the azimuth, and z
the height. However, the radius is also often denoted r, the
azimuth by θ or t, and the third coordinate by h or (if the
cylindrical axis is considered horizontal) x, or any contextspecific letter.
Coordinate system conversions
The cylindrical coordinate system is one of many threedimensional coordinate systems. The following formulae may
be used to convert between them.
Cartesian coordinates
For the conversion between cylindrical and Cartesian
coordinate co-ordinates, it is convenient to assume that the
reference plane of the former is the Cartesian x–y plane (with
equation z = 0), and the cylindrical axis is the Cartesian z axis.
Then the z coordinate is the same in both systems, and the
correspondence between cylindrical (ρ,φ) and Cartesian (x,y)
are the same as for polar coordinates, namely
x   cos 
y   sin 
The coordinate surfaces of the cylindrical coordinates
(ρ, φ, z). The red cylinder shows the points with ρ=2, the blue
in one direction, and  
x
2
 y2

The arcsin function is the inverse of the sine function, and is
assumed to return an angle in the range [−π/2,+π/2] =
[−90°,+90°]. These formulas yield an azimuth φ in the range
[−90°,+270°].
Spherical coordinates
Spherical coordinates (radius r, elevation or inclination
θ, azimuth φ), may be converted into cylindrical coordinates
by:
θ is elevation:
θ is inclination:
  r cos 
  r sin 
 
 
z  r sin 
z  r cos
Spherical coordinates, also called spherical polar coordinates
(Walton 1967, Arfken 1985), are a system of curvilinear
coordinates that are natural for describing positions on a sphere
or spheroid. Define ѳ to be the azimuthal angle in the x-y plane
from the x-axis with 0  2 (denoted when referred to as
the longitude),  to be the polar angle (also known as the
LECTURE NO.:04 SPHERICAL COORDINATE
SYSTEM
zenith angle and colatitude, with   90 0   where  is the
latitude) from the positive z-axis with 0     , and r to be
A spherical coordinate system is a coordinate system
for three-dimensional space where the position of a point is
specified by three numbers: the radial distance of that point
from a fixed origin, its polar angle measured from a fixed
zenith direction, and the azimuth angle of its orthogonal
projection on a reference plane that passes through the origin
and is orthogonal to the zenith, measured from a fixed
reference direction on that plane.
distance (radius) from a point to the origin. This is the
convention commonly used in mathematics.
In this work, following the mathematics convention, the
symbols for the radial, azimuth, and zenith angle coordinates
are taken as r,  , and  , respectively. Note that this definition
provides a logical extension of the usual polar coordinates
notation, with  remaining the angle in the xy plane and
 becoming the angle out of that plane.
The radial distance is also called the radius or radial
coordinate. The polar angle may be called co-latitude, zenith
angle, normal angle, or inclination angle.
The use of symbols and the order of the coordinates differs
between sources. In one system frequently encountered in
physics (r, θ, φ) gives the radial distance, polar angle, and
azimuthal angle, whereas in another system used in many
mathematics books (r, θ, φ) gives the radial distance, azimuthal
angle, and polar angle. In both systems ρ is often used instead
of r. Other conventions are also used, so great care needs to be
taken to check which one is being used.
Unique coordinates
Spherical coordinates (r, θ, φ) as commonly used in physics:
radial distance r, polar angle θ (theta), and azimuthal angle φ
(phi). The symbol ρ (rho) is often used instead of r.
Any spherical coordinate triplet (r, θ, φ) specifies a
single point of three-dimensional space. On the other hand,
every point has infinitely many equivalent spherical
coordinates. One can add or subtract any number of full turns
to either angular measure without changing the angles
themselves, and therefore without changing the point. It is also
convenient, in many contexts, to allow negative radial
distances, with the convention that (−r, θ, φ) is equivalent to (r,
θ + 180°, φ) for any r, θ, and φ. Moreover, (r, −θ, φ) is
equivalent to (r, θ, φ + 180°).
If it is necessary to define a unique set of spherical coordinates
for each point, one may restrict their ranges. A common choice
is:
Spherical coordinates (r, θ, φ) as often used in mathematics:
radial distance r, azimuthal angle θ, and polar angle φ. The
meanings of θ and φ have been swapped compared to the
physics convention.
r≥0
0° ≤ θ ≤ 180° (π rad)
0° ≤ φ < 360° (2π rad)
However, the azimuth φ is often restricted to the interval
(−180°, +180°], or (−π, +π] in radians, instead of [0, 360°).
This is the standard convention for geographic longitude.
The range [0°, 180°] for inclination is equivalent to [−90°,
+90°] for elevation (latitude).
Even with these restrictions, if θ is zero or 180° (elevation is
90° or −90°) then the azimuth angle is arbitrary; and if r is
zero, both azimuth and inclination/elevation are arbitrary. To
make the coordinates unique, one can use the convention that
in these cases the arbitrary coordinates are zero.
 y
x
  arctan  
Conversely, the Cartesian coordinates may be retrieved from
the spherical coordinates (radius r, inclination θ, azimuth φ),
where r ∈ [0, ∞), θ ∈ [0, π], φ ∈ [0, 2π), by:
x  r sin  cos 
y  r sin  sin 
z  r cos 
Coordinate system conversions
Cylindrical coordinates
As the spherical coordinate system is only one of many threedimensional coordinate systems, there exist equations for
converting coordinates between the spherical coordinate
system and others.
Cylindrical coordinates (radius ρ, azimuth φ, elevation z) may
be converted into spherical coordinates (radius r, inclination θ,
azimuth φ), by the formulas
Cartesian coordinates
The spherical coordinates of a point in the ISO convention
(radius r, inclination θ, azimuth φ) can be obtained from its
Cartesian coordinates (x, y, z) by the formulae

r  x2  y2  z2

  arccos


x
 y2  z2


  arctan   z   arccos






2
2 
 z 

z

 

z
2

r   2  z2





Conversely, the spherical coordinates may be converted into
cylindrical coordinates by the formulae
  r sin 
 
z  r cos 
LECTURE NO.06
Differential Elements of Length,
Surface, and Volume
Rectangular coordinate system
A differential volume element in the rectangular coordinate
system is generated by making differential changes dx, dy, and
dz along the unit vectors x, y and z, respectively, as
illustrated in Figure 2.18a. The differential volume is given by
the expression
The general differential length element from P to Q is
Cylindrical coordinate system
The differential surfaces in the positive direction of the unit
vectors are
Figure above shows differential elements in a rectangular
coordinate system
The volume is enclosed by six differential surfaces. Each
surface is defined by a unit vector normal to that surface. Thus,
we can express the differential surfaces in the direction of
positive unit vectors (see Figure 2.18b) as
Figure: Differential elements in a cylindrical coordinate system
The differential length vector from P to Q is
The surface element in a surface of azimuth
vertical half-plane) is
Spherical coordinate system
The volume element spanning from to
and to
is
LECTURE NO.07
constant (a
, to
,
Electric field due to Continuous charge
Distributions
Point Charge: Charge whose volume is very small when compared to
the distance of separation under consideration.
Line Charge: Charge whose surface area is very small when
compared to its length.
The surface element spanning from to
and
on a spherical surface at (constant) radius is
to
Thus the differential solid angle is
Surface Charge: Charge whose thickness is very small when
compared to its area.
Volume Charge: A charged object comprising of three dimensions.
Coulomb’s law can be applied to find the force exerted on some
point charge q due to line, surface & volume distribution of charges.
Here the charges are not concentrated rather they are distributed.
In this situation the total sum must be replaced by integral since a
continuous distribution of charges are taken care of.
Electric field due to Continuous Uniform charge Distributions
The surface element in a surface of polar angle
cone with vertex the origin) is
constant (a
The charge density (  L in C/m), surface charge density (  S in
C/m2), volume charge density (  V in C/m3)
Line charge dQ   L dl
of conductor & not in their interiors, hence  S is known as surface
Surface charge dQ   S ds
charge density.
Volume charge dQ   v dv
LECTURE NO.08
  S dS a (Surface charge)
Hence Electric field intensity E 
R
4 0 R 2
E

dV 
a R (Volume charge)
40 R 2
Del operator
V
Volume Charge density (  V )
The space between the control grid and thee cathode in the
electron gun assembly of a cathode ray tube operating with space
charge, we can replace this distribution of very small particles with a
smooth continuous distribution described by a volume charge
density and its unit is Coulombs/m3.
Line charge density (  L )
If a filament like distribution of volume charge is considered such as
a very fine, sharp beam in a cathode ray tube of very small radius it
is convenient to heat the charge as a line charge density. its unit is
Coulombs/m.
Surface charge density (  S )
One basic charge configuration often used to approximate that
found on the conductors of a strip transmission line or parallel plate
capacitor is the infinite sheet of charge having a uniform charge
density of  S Coulomb/m2.The static charges reside on the surface
Del operator,represented bythe nabla symbol
Del, or nabla, is an operator used in mathematics, in particular,
in vector calculus, as a vector differential operator, usually
represented by the nabla symbol ∇. When applied to a function
defined on a one-dimensional domain, it denotes its standard
derivative as defined in calculus. When applied to a field (a
function defined on a multi-dimensional domain), del may
denote the gradient (locally steepest slope) of a scalar field (or
sometimes of a vector field, as in the Navier–Stokes
equations), the divergence of a vector field, or the curl
(rotation) of a vector field, depending on the way it is applied.
Strictly speaking, del is not a specific operator, but rather a
convenient mathematical notation for those three operators,
that makes many equations easier to write and remember. The
del symbol can be interpreted as a vector of partial derivative
operators, and its three possible meanings—gradient,
divergence, and curl—can be formally viewed as the product of
scalars, dot product, and cross product, respectively, of the del
"operator" with the field. These formal products do not
necessarily commute with other operators or products.
Definition
In the Cartesian coordinate system Rn with coordinates
and standard basis
defined in terms of partial derivative operators as
, del is
In three-dimensional Cartesian coordinate system R3 with
coordinates
written as
and standard basis
, del is
It always points in the direction of greatest increase of f, and it
has a magnitude equal to the maximum rate of increase at the
point—just like a standard derivative. In particular, if a hill is
defined as a height function over a plane h(x,y), the 2d
projection of the gradient at a given location will be a vector in
the xy-plane (visualizable as an arrow on a map) pointing along
the steepest direction. The magnitude of the gradient is the
value of this steepest slope.
In particular, this notation is powerful because the gradient
product rule looks very similar to the 1d-derivative case:
However, the rules for dot products do not turn out to be
simple, as illustrated by:
Del can also be expressed in other coordinate systems, see for
example del in cylindrical and spherical coordinates.
Notational uses
Divergence
Del is used as a shorthand form to simplify many long
mathematical expressions. It is most commonly used to
simplify expressions for the gradient, divergence, curl,
directional derivative, and Laplacian.
The
divergence
of
a
vector
field
is a scalar function that
can be represented as:
Gradient
The vector derivative of a scalar field f is called the gradient,
and it can be represented as:
The divergence is roughly a measure of a vector field's increase
in the direction it points; but more accurately, it is a measure of
that field's tendency to converge toward or repel from a point.
The power of the del notation is shown by the following
product rule:
The formula for the vector product is slightly less intuitive,
because this product is not commutative:
Directional derivative
The directional derivative of a scalar field f(x,y,z) in the
direction
is defined as:
Curl
The
curl
of
a
vector
field
is a vector function that
can be represented as:
The curl at a point is proportional to the on-axis torque to
which a tiny pinwheel would be subjected if it were centered at
that point.
The vector product operation can be visualized as a pseudodeterminant:
This gives the change of a field f in the direction of a. In
operator notation, the element in parentheses can be considered
a single coherent unit; fluid dynamics uses this convention
extensively, terming it the convective derivative—the
"moving" derivative of the fluid.
Laplacian
The Laplace operator is a scalar operator that can be applied to
either vector or scalar fields; for cartesian coordinate systems it
is defined as:
and the definition for more general coordinate systems is given
in Vector Laplacian.
Again the power of the notation is shown by the product rule:
)
Unfortunately the rule for the vector product does not turn out
to be simple:
The Laplacian is ubiquitous throughout modern
mathematical physics, appearing in Laplace's equation,
Poisson's equation, the heat equation, the wave equation, and
the Schrödinger equation—to name a few.
Second derivatives
two of them are always equal:
DCG chart: A simple chart depicting all rules pertaining
to second derivatives. D, C, G, L and CC stand for divergence,
curl, gradient, Laplacian and curl of curl, respectively. Arrows
indicate existence of second derivatives. Blue circle in the
middle represents curl of curl, whereas the other two red circles
(dashed) mean that DD and GG do not exist.
When del operates on a scalar or vector, either a scalar
or vector is returned. Because of the diversity of vector
products (scalar, dot, cross) one application of del already
gives rise to three major derivatives: the gradient (scalar
product), divergence (dot product), and curl (cross product).
Applying these three sorts of derivatives again to each other
gives five possible second derivatives, for a scalar field f or a
vector field v; the use of the scalar Laplacian and vector
Laplacian gives two more:
These are of interest principally because they are not always
unique or independent of each other. As long as the functions
are well-behaved, two of them are always zero:
The 3 remaining vector derivatives are related by the equation:
And one of them can even be expressed with the tensor
product, if the functions are well-behaved:
LECTURE NO.:09
GRADIENT OF A SCALAR
The gradient of the function f(x,y) = −(cos2x + cos2y)2
depicted as a projected vector field on the bottom plane.
The gradient (or gradient vector field) of a scalar
function f(x1, x2, x3, ..., xn) is denoted ∇f or
where ∇
(the nabla symbol) denotes the vector differential
operator, del. The notation "grad(f)" is also commonly
used for the gradient. The gradient of f is defined as the
unique vector field whose dot product with any vector v
at each point x is the directional derivative of f along v.
That is,
is:
In some applications it is customary to represent the
gradient as a row vector or column vector of its
components in a rectangular coordinate system.
In a rectangular coordinate system, the gradient is the
vector field whose components are the partial derivatives
of f:
where the ei are the orthogonal unit vectors pointing in
the coordinate directions. When a function also depends
on a parameter such as time, the gradient often refers
simply to the vector of its spatial derivatives only.
In the three-dimensional Cartesian coordinate system,
this is given by
where i, j, k are the standard unit vectors. For example,
the gradient of the function
In the above two images, the values of the function are
represented in black and white, black representing higher
values, and its corresponding gradient is represented by
blue arrows.
In mathematics, the gradient is a generalization of the
usual concept of derivative of a function in one
dimension to a function in several dimensions. If f(x1, ...,
xn) is a differentiable, scalar-valued function of standard
Cartesian coordinates in Euclidean space, its gradient is
the vector whose components are the n partial derivatives
of f. It is thus a vector-valued function.
Similarly to the usual derivative, the gradient represents
the slope of the tangent of the graph of the function.
More precisely, the gradient points in the direction of the
greatest rate of increase of the function and its magnitude
is the slope of the graph in that direction. The
components of the gradient in coordinates are the
coefficients of the variables in the equation of the tangent
space to the graph. This characterizing property of the
gradient allows it to be defined independently of a choice
of coordinate system, as a vector field whose components
in a coordinate system will transform when going from
one coordinate system to another.
The Jacobian is the generalization of the gradient for
vector-valued functions of several variables and
differentiable maps between Euclidean spaces or, more
generally, manifolds. A further generalization for a
function between Banach spaces is the Fréchet
derivative.
Linear approximation to a function
The gradient of a function f from the Euclidean space ℝn
to ℝ at any particular point x0 in ℝn characterizes the best
linear approximation to f at x0. The approximation is as
follows:
for x close to x0, where
is the gradient of f
computed at x0, and the dot denotes the dot product on
ℝn. This equation is equivalent to the first two terms in
the multi-variable Taylor Series expansion of f at x0.
Gradient as a derivative
Let U be an open set in Rn. If the function f : U → R is
differentiable, then the differential of f is the (Fréchet)
derivative of f. Thus ∇f is a function from U to the space
R such that
where ⋅ is the dot product.
As a consequence, the usual properties of the derivative
hold for the gradient:
Linearity
The gradient is linear in the sense that if f and g are two
real-valued functions differentiable at the point a ∈ Rn,
and α and β are two constants, then αf + βg is
differentiable at a, and moreover
Product rule
If f and g are real-valued functions differentiable at a
point a ∈ Rn, then the product rule asserts that the product
(fg)(x) = f(x)g(x) of the functions f and g is differentiable
at a, and
Chain rule
Suppose that f : A → R is a real-valued function defined
on a subset A of Rn, and that f is differentiable at a point
a. There are two forms of the chain rule applying to the
gradient. First, suppose that the function g is a parametric
curve; that is, a function g : I → Rn maps a subset I ⊂ R
into Rn. If g is differentiable at a point c ∈ I such that g(c)
= a, then
where ∘ is the composition operator : (g ∘ f )(x) = g(f(x)).
More generally, if instead I ⊂ Rk, then the following
holds:
More generally, any embedded hypersurface in a
Riemannian manifold can be cut out by an equation of
the form F(P) = 0 such that dF is nowhere zero. The
gradient of F is then normal to the hypersurface.
Similarly, an affine algebraic hypersurface may be
defined by an equation F(x1, ..., xn) = 0, where F is a
polynomial. The gradient of F is zero at a singular point
of the hypersurface (this is the definition of a singular
point). At a non-singular point, it is a nonzero normal
vector.
Cylindrical and spherical coordinates
where (Dg)T denotes the transpose Jacobian matrix.
For the second form of the chain rule, suppose that h : I
→ R is a real valued function on a subset I of R, and that
h is differentiable at the point f(a) ∈ I. Then
In cylindrical coordinates, the gradient is given by
(Schey 1992, pp. 139–142):
Further properties and applications
where ϕ is the azimuthal angle, z is the axial coordinate,
and eρ, eφ and ez are unit vectors pointing along the
coordinate directions.
A level surface, or isosurface, is the set of all points
where some function has a given value.
In spherical coordinates (Schey 1992, pp. 139–142):
If f is differentiable, then the dot product (∇f)x ⋅ v of the
gradient at a point x with a vector v gives the directional
derivative of f at x in the direction v. It follows that in
this case the gradient of f is orthogonal to the level sets of
f. For example, a level surface in three-dimensional space
is defined by an equation of the form F(x, y, z) = c. The
gradient of F is then normal to the surface.
where ϕ is the azimuth angle and θ is the zenith angle.
For the gradient in other orthogonal coordinate systems,
see Orthogonal coordinates (Differential operators in
three dimensions).
Gradient of a vector
FIELD
In rectangular coordinates, the gradient of a vector field f
= (f1, f2, f3) is defined by
In physical terms, the divergence of a three-dimensional
vector field is the extent to which the vector field flow
behaves like a source or a sink at a given point. It is a
local measure of its "outgoingness"—the extent to which
there is more exiting an infinitesimal region of space than
entering it. If the divergence is nonzero at some point
then there must be a source or sink at that position. (Note
that we are imagining the vector field to be like the
velocity vector field of a fluid (in motion) when we use
the terms flow, sink and so on.)
where the Einstein summation notation is used and the
product of the vectors ei, ek is a tensor of type (2,0), or
the Jacobian matrix
.
In curvilinear coordinates, or more generally on a curved
manifold, the gradient involves Christoffel symbols:
where gjk are the components of the metric tensor and the
ei are the coordinate vectors.
Expressed more invariantly, the gradient of a vector field
f can be defined by the Levi-Civita connection and
metric tensor:[1]
where
is the connection.
LECTURE NO.:10 DIVERGENCE OF A VECTOR
More rigorously, the divergence of a vector field F at a
point p is defined as the limit of the net flow of F across
the smooth boundary of a three-dimensional region V
divided by the volume of V as V shrinks to p. Formally,
where |V | is the volume of V, S(V) is the boundary of V,
and the integral is a surface integral with n being the
outward unit normal to that surface. The result, div F, is a
function of p. From this definition it also becomes
explicitly visible that div F can be seen as the source
density of the flux of F.
In light of the physical interpretation, a vector field with
constant zero divergence is called incompressible or
solenoidal – in this case, no net flow can occur across
any closed surface.
The intuition that the sum of all sources minus the sum of
all sinks should give the net flow outwards of a region is
made precise by the divergence theorem.
Application in Cartesian coordinates
Let x, y, z be a system of Cartesian coordinates in 3dimensional Euclidean space, and let i, j, k be the
corresponding basis of unit vectors.
The divergence of a continuously differentiable vector
field F = U i + V j + W k is equal to the scalar-valued
function:
Cylindrical coordinates
For a vector expressed in cylindrical coordinates as
where ea is the unit vector in direction a, the divergence is
Although expressed in terms of coordinates, the result is
invariant under orthogonal transformations, as the
physical interpretation suggests.
The common notation for the divergence ∇ · F is a
convenient mnemonic, where the dot denotes an
operation reminiscent of the dot product: take the
components of ∇ (see del), apply them to the components
of F, and sum the results. Because applying an operator is
different from multiplying the components, this is
considered an abuse of notation.
The divergence of a continuously differentiable secondorder tensor field is a first-order tensor field:
Spherical coordinates
In spherical coordinates, with the angle with the z axis
and the rotation around the z axis, the divergence reads
LECTURE NO.11 DIVERGENCE THEOREM
In vector calculus, the divergence theorem, also known
as Gauss's theorem or Ostrogradsky's theorem, is a result
that relates the flow (that is, flux) of a vector field
through a surface to the behavior of the vector field
inside the surface.
More precisely, the divergence theorem states that the
outward flux of a vector field through a closed surface is
equal to the volume integral of the divergence over the
region inside the surface. Intuitively, it states that the
sum of all sources minus the sum of all sinks gives the
net flow out of a region.
The divergence theorem is an important result for the
mathematics of engineering, in particular in electrostatics
and fluid dynamics.
In physics and engineering, the divergence theorem is
usually applied in three dimensions. However, it
generalizes to any number of dimensions. In one
dimension, it is equivalent to the fundamental theorem of
calculus. In two dimensions, it is equivalent to Green's
theorem.
The divergence theorem can be used to calculate a flux
through a closed surface that fully encloses a volume,
like any of the surfaces on the left. It can not directly be
used to calculate the flux through surfaces with
boundaries, like those on the right. (Surfaces are blue,
boundaries are red.)
The theorem is a special case of the more general Stokes'
theorem
Suppose V is a subset of
(in the case of n = 3, V
represents a volume in 3D space) which is compact and
has a piecewise smooth boundary S (also indicated with
∂V = S ). If F is a continuously differentiable vector field
defined on a neighborhood of V, then we have:[6]
Mathematical statement
A region V bounded by the surface S = ∂V with the
surface normal n
The left side is a volume integral over the volume V, the
right side is the surface integral over the boundary of the
volume V. The closed manifold ∂V is quite generally the
boundary of V oriented by outward-pointing normals,
and n is the outward pointing unit normal field of the
boundary ∂V. (dS may be used as a shorthand for ndS.)
The symbol within the two integrals stresses once more
that ∂V is a closed surface. In terms of the intuitive
description above, the left-hand side of the equation
represents the total of the sources in the volume V, and
the right-hand side represents the total flow across the
boundary ∂V.

Applying the divergence theorem to the crossproduct of a vector field F and a non-zero
constant vector c, the following theorem can be
proven:[7]
Corollaries
By applying the divergence theorem in various contexts,
other useful identities can be derived

Applying the divergence theorem to the product
of a scalar function g and a vector field F, the
result is
A special case of this is F = ∇ f , in which case the
theorem is the basis for Green's identities.

Applying the divergence theorem to the crossproduct of two vector fields F × G, the result is
Example

Applying the divergence theorem to the product
of a scalar function, f , and a non-zero constant
vector c, the following theorem can be proven:[7]
The vector field corresponding to the example shown.
Note, vectors may point into or out of the sphere.
Suppose we wish to evaluate
where S is the unit sphere defined by
and F is the vector field
LECTURE NO.12 CURL OF A VECTOR
In vector calculus, the curl is a vector operator that
describes the infinitesimal rotation of a 3-dimensional
vector field. At every point in the field, the curl of that
point is represented by a vector. The attributes of this
vector (length and direction) characterize the rotation at
that point.
The direction of the curl is the axis of rotation, as
determined by the right-hand rule, and the magnitude of
the curl is the magnitude of rotation. If the vector field
represents the flow velocity of a moving fluid, then the
curl is the circulation density of the fluid. A vector field
whose curl is zero is called irrotational. The curl is a
form of differentiation for vector fields. The
corresponding form of the fundamental theorem of
calculus is Stokes' theorem, which relates the surface
integral of the curl of a vector field to the line integral of
the vector field around the boundary curve.
The alternative terminology rotor or rotational and
alternative notations rot F and ∇ × F are often used (the
former especially in many European countries, the latter,
using the del operator and the cross product, is more used
in other countries) for curl and curl F.
Unlike the gradient and divergence, curl does not
generalize as simply to other dimensions; some
generalizations are possible, but only in three dimensions
is the geometrically defined curl of a vector field again a
vector field. This is a similar phenomenon as in the 3
dimensional cross product, and the connection is
reflected in the notation ∇ × for the curl.
The name "curl" was first suggested by James Clerk
Maxwell in 1871[1] but the concept was apparently first
used in the construction of an optical field theory by
James MacCullagh in 1839.
Definition
Convention for vector orientation of the line integral
Implicitly, curl is defined by:
The components of F at position r, normal and tangent to
a closed curve C in a plane, enclosing a planar vector
area A = An.
The curl of a vector field F, denoted by curl F, or ∇ × F,
or rot F, at a point is defined in terms of its projection
onto various lines through the point. If is any unit
vector, the projection of the curl of F onto is defined to
be the limiting value of a closed line integral in a plane
orthogonal to as the path used in the integral becomes
infinitesimally close to the point, divided by the area
enclosed.
As such, the curl operator maps continuously
differentiable functions f : R3 → R3 to continuous
functions g : R3 → R3. In fact, it maps Ck functions in R3
to Ck-1 functions in R3.
where
is a line integral along the boundary of the
area in question, and |A| is the magnitude of the area. If
is an outward pointing in-plane normal, whereas is the
unit vector perpendicular to the plane (see caption at
right), then the orientation of C is chosen so that a
tangent vector to C is positively oriented if and only if
forms a positively oriented basis for R3 (righthand rule).
The above formula means that the curl of a vector field is
defined as the infinitesimal area density of the circulation
of that field. To this definition fit naturally

the Kelvin-Stokes theorem, as a global formula
corresponding to the definition, and the following
"easy to memorize" definition of the curl in
curvilinear orthogonal coordinates, e.g. in
Cartesian coordinates, spherical, cylindrical, or
even elliptical or parabolical coordinates:
representations have been derived.
The notation ∇ × F has its origins in the similarities to the
3 dimensional cross product, and it is useful as a
mnemonic in Cartesian coordinates if ∇ is taken as a
vector differential operator del. Such notation involving
operators is common in physics and algebra. However, in
certain coordinate systems, such as polar-toroidal
coordinates (common in plasma physics), using the
notation ∇ × F will yield an incorrect result.
,
,
.
Note
that
the
equation
for
each
component,
can be obtained by exchanging each
occurrence of a subscript 1, 2, 3 in cyclic permutation:
1→2, 2→3, and 3→1 (where the subscripts represent the
relevant indices).
Expanded in Cartesian coordinates (see Del in cylindrical
and spherical coordinates for spherical and cylindrical
coordinate representations), ∇ × F is, for F composed of
[Fx, Fy, Fz]:
If (x1, x2, x3) are the Cartesian coordinates and (u1,u2,u3)
are the orthogonal coordinates, then
where i, j, and k are the unit vectors for the x-, y-, and zaxes, respectively. This expands as follows:[6]
is the length of the coordinate vector corresponding to ui.
The remaining two components of curl result from cyclic
permutation of indices: 3,1,2 → 1,2,3 → 2,3,1.
Usage
In practice, the above definition is rarely used because in
virtually all cases, the curl operator can be applied using
some set of curvilinear coordinates, for which simpler
Although expressed in terms of coordinates, the result is
invariant under proper rotations of the coordinate axes
but the result inverts under reflection.
In a general coordinate system, the curl is given by
where ε denotes the Levi-Civita symbol, the metric
tensor is used to lower the index on F, and the Einstein
summation convention implies that repeated indices are
summed over. Equivalently,
where ek are the coordinate vector fields. Equivalently,
using the exterior derivative, the curl can be expressed
as:
Here and are the musical isomorphisms, and is the
Hodge dual. This formula shows how to calculate the
curl of F in any coordinate system, and how to extend the
curl to any oriented three-dimensional Riemannian
manifold. Since this depends on a choice of orientation,
curl is a chiral operation. In other words, if the
orientation is reversed, then the direction of the curl is
also reversed.
Simply by visual inspection, we can see that the field is
rotating. If we place a paddle wheel anywhere, we see
immediately its tendency to rotate clockwise. Using the
right-hand rule, we expect the curl to be into the page. If
we are to keep a right-handed coordinate system, into the
page will be in the negative z direction. The lack of x and
y directions is analogous to the cross product operation.
If we calculate the curl:
Examples
A simple vector field
Take the vector field, which depends on x and y linearly:
Its plot looks like this:
Which is indeed in the negative z direction, as expected.
In this case, the curl is actually a constant, irrespective of
position. The "amount" of rotation in the above vector
field is the same at any point (x, y). Plotting the curl of F
is not very interesting:
A more involved example
Suppose we now consider a slightly more complicated
vector field:
Its plot:
We might not see any rotation initially, but if we closely
look at the right, we see a larger field at, say, x=4 than at
x=3. Intuitively, if we placed a small paddle wheel there,
the larger "current" on its right side would cause the
paddlewheel to rotate clockwise, which corresponds to a
curl in the negative z direction. By contrast, if we look at
a point on the left and placed a small paddle wheel there,
the larger "current" on its left side would cause the
paddlewheel to rotate counterclockwise, which
corresponds to a curl in the positive z direction. Let's
check out our guess by doing the math:
Indeed the curl is in the positive z direction for negative
x and in the negative z direction for positive x, as
expected. Since this curl is not the same at every point,
its plot is a bit more interesting:
We note that the plot of this curl has no dependence on y
or z (as it shouldn't) and is in the negative z direction for
positive x and in the positive z direction for negative x.
If φ is a scalar valued function and F is a vector field,
then
Identities
Consider the example ∇ × (v × F). Using Cartesian
coordinates, it can be shown that
In the case where the vector field v and ∇ are
interchanged:
which introduces the Feynman subscript notation ∇F,
which means the subscripted gradient operates only on
the factor F.
Descriptive examples


In a vector field describing the linear velocities of
each part of a rotating disk, the curl has the same
value at all points.
Of the four Maxwell's equations, two—Faraday's
law and Ampère's law—can be compactly
expressed using curl. Faraday's law states that the
curl of an electric field is equal to the opposite of
the time rate of change of the magnetic field,
while Ampère's law relates the curl of the
magnetic field to the current and rate of change of
the electric field.
Generalizations
Another example is ∇ × (∇ × F). Using Cartesian
coordinates, it can be shown that:
which can be construed as a special case of the previous
example with the substitution v → ∇.
(Note: ∇2F represents the vector Laplacian of F)
The curl of the gradient of any scalar field φ is always the
zero vector:
The vector calculus operations of grad, curl, and div are
most easily generalized and understood in the context of
differential forms, which involves a number of steps. In a
nutshell, they correspond to the derivatives of 0-forms, 1forms, and 2-forms, respectively. The geometric
interpretation of curl as rotation corresponds to
identifying bivectors (2-vectors) in 3 dimensions with the
special orthogonal Lie algebra so(3) of infinitesimal
rotations (in coordinates, skew-symmetric 3 × 3
matrices), while representing rotations by vectors
corresponds to identifying 1-vectors (equivalently, 2vectors) and so(3), these all being 3-dimensional spaces.
Differential
In 3 dimensions, a differential 0-form is simply a
function f(x, y, z); a differential 1-form is the following
expression:
a differential 2form
is
the
formal
sum:
and a
differential 3-form is defined by a single term:
(Here the a-coefficients are real
functions; the "wedge products", e.g.
can be
interpreted as some kind of oriented area elements,
, etc.) The exterior derivative
of a k-form in R3 is defined as the (k+1)-form from above
(and in Rn if, e.g.,
then the exterior derivative d leads to
The exterior derivative of a 1-form is therefore a 2-form,
and that of a 2-form is a 3-form. On the other hand,
because of the interchangeability of mixed derivatives,
e.g. because of
the twofold application of the exterior derivative leads to
0.
Thus, denoting the space of k-forms by
exterior derivative by d one gets a sequence:
Here
algebra
and the
is the space of sections of the exterior
vector
bundle
over
dimension is the binomial coefficient
Rn,
whose
note that
for k > 3 or k < 0. Writing only
dimensions, one obtains a row of Pascal's triangle:
0 → 1 → 3 → 3 → 1 → 0;
the 1-dimensional fibers correspond to functions, and the
3-dimensional fibers to vector fields, as described below.
Note that modulo suitable identifications, the three
nontrivial occurrences of the exterior derivative
correspond to grad, curl, and div.
Differential forms and the differential can be defined on
any Euclidean space, or indeed any manifold, without
any notion of a Riemannian metric. On a Riemannian
manifold, or more generally pseudo-Riemannian
manifold, k-forms can be identified with k-vector fields
(k-forms are k-covector fields, and a pseudo-Riemannian
metric gives an isomorphism between vectors and
covectors), and on an oriented vector space with a
nondegenerate form (an isomorphism between vectors
and covectors), there is an isomorphism between kvectors and (n−k)-vectors; in particular on (the tangent
space of) an oriented pseudo-Riemannian manifold. Thus
on an oriented pseudo-Riemannian manifold, one can
interchange k-forms, k-vector fields, (n−k)-forms, and
(n−k)-vector fields; this is known as Hodge duality.
Concretely, on R3 this is given by:


1-forms
and
1-vector
fields: the 1-form
corresponds to the
vector field
1-forms and 2-forms: one replaces dx by the
"dual" quantity
(i.e., omit dx), and
likewise, taking care of orientation: dy
corresponds to
and dz
corresponds to
Thus the form
corresponds to the
"dual
form"
Thus, identifying 0-forms and 3-forms with functions,
and 1-forms and 2-forms with vector fields:



grad takes a function (0-form) to a vector field (1form);
curl takes a vector field (1-form) to a vector field
(2-form);
div takes a vector field (2-form) to a function (3form)
On the other hand the fact that d2 = 0 corresponds to the
identities curl grad f = 0 and
for any
function f or vector field
Grad and div generalize to all oriented pseudoRiemannian manifolds, with the same geometric
interpretation, because the spaces of 0-forms and n-forms
is always (fiberwise) 1-dimensional and can be identified
with scalar functions, while the spaces of 1-forms and
(n−1)-forms are always fiberwise n-dimensional and can
be identified with vector fields.
Curl does not generalize in this way to 4 or more
dimensions (or down to 2 or fewer dimensions); in 4
dimensions the dimensions are
0 → 1 → 4 → 6 → 4 → 1 → 0;
so the curl of a 1-vector field (fiberwise 4-dimensional)
is a 2-vector field, which is fiberwise 6-dimensional, one
has
which yields a sum of six independent terms, and cannot
be identified with a 1-vector field. Nor can one
meaningfully go from a 1-vector field to a 2-vector field
to a 3-vector field (4 → 6 → 4), as taking the differential
twice yields zero (d2 = 0). Thus there is no curl function
from vector fields to vector fields in other dimensions
arising in this way.
However, one can define a curl of a vector field as a 2vector field in general, as described below.
LECTURE NO.13 LAPLACIAN OF A SCALAR
The Laplacian for a scalar function
differential operator defined by
is a scalar
generalization to a tensor Laplacian.
where the are the scale factors of the coordinate .Note
that the operator
is commonly written as by
mathematicians
The Laplacian is extremely important in mechanics,
electromagnetics, wave theory, and quantum mechanics,
and appears in Laplace's equation
The following table gives the form of the Laplacian in
several common coordinate systems.
coordinate system
Cartesian
coordinates
cylindrical
coordinates
the Helmholtz differential equation
the wave equation
parabolic
coordinates
parabolic
cylindrical
coordinates
spherical
coordinates
and the Schrödinger equation
The finite difference form is
The analogous operator obtained by generalizing from
three dimensions to four-dimensional spacetime is
denoted and is known as the d'Alembertian. A version
of the Laplacian that operates on vector functions is
known as the vector Laplacian, and a tensor Laplacian
can be similarly defined. The square of the Laplacian
is known as the biharmonic operator.
A vector Laplacian can also be defined, as can its
For a pure radial function
,
Using the vector derivative identity
electrostatic force of interaction between two point
charges is directly proportional to the scalar
multiplication of the magnitudes of charges and inversely
proportional to the square of the distance between
them.[12]
so
The force is along the straight line joining them.
If the two charges have the same sign, the
electrostatic force between them is repulsive; if
they have different sign, the force between them
is attractive.
Therefore, for a radial power law,
Coulomb's law can also be stated as a simple
mathematical expression. The scalar and vector forms of
the mathematical equation are
and
LECTURE NO.14 Coulomb's law
Coulomb's law, or Coulomb's inverse-square law, is a
law of physics describing the electrostatic interaction
between electrically charged particles. The law was first
published in 1785 by French physicist Charles Augustin
de Coulomb and was essential to the development of the
theory of electromagnetism. It is analogous to Isaac
Newton's inverse-square law of universal gravitation.
Coulomb's law can be used to derive Gauss's law, and
vice versa. The law has been tested heavily, and all
observations have upheld the law's principle.
Coulomb's law states that:The magnitude of the
respectively,
where
is
Coulomb's
constant
(
),
and are the signed magnitudes of the charges, the
scalar is the distance between the charges, the vector
is the vectorial distance between the
charges, and
(a unit vector pointing
from to ). The vector form of the equation calculates
the force
applied on by . If
is used instead,
then the effect on can be found. It can be also
calculated using Newton's third law:
.
Units
Electromagnetic theory is usually expressed using the
standard SI units. Force is measured in newtons, charge
in coulombs, and distance in metres. Coulomb's constant
is given by
. The constant
is the
2
−2
−1
permittivity of free space in C m N . And is the
relative permittivity of the material in which the charges
are immersed, and is dimensionless.
The SI derived units for the electric field are volts per
meter, newtons per coulomb, or tesla meters per second.
Coulomb's law and Coulomb's constant can also be
interpreted in various terms:


Electric field
Atomic units. In atomic units the force is
expressed in hartrees per Bohr radius, the
charge in terms of the elementary charge,
and the distances in terms of the Bohr
radius.
Electrostatic units or Gaussian units. In
electrostatic units and Gaussian units, the
unit charge (esu or statcoulomb) is defined
in such a way that the Coulomb constant k
disappears because it has the value of one
and becomes dimensionless.
If the two charges have the same sign, the electrostatic
force between them is repulsive; if they have different
sign, the force between them is attractive.
An electric field is a vector field that associates to each
point in space the Coulomb force experienced by a test
charge. In the simplest case, the field is considered to be
generated solely by a single source point charge. The
strength and direction of the Coulomb force on a test
charge depends on the electric field
that it finds
itself in, such that
. If the field is generated by
a positive source point charge , the direction of the
electric field points along lines directed radially outwards
from it, i.e. in the direction that a positive point test
charge
would move if placed in the field. For a
negative point source charge, the direction is radially
inwards.
The magnitude of the electric field
can be derived
from Coulomb's law. By choosing one of the point
charges to be the source, and the other to be the test
charge, it follows from Coulomb's law that the magnitude
of the electric field
created by a single source point
charge at a certain distance from it in vacuum is given
by:
.
Coulomb's constant
Coulomb's constant is a proportionality factor that
appears in Coulomb's law as well as in other electricrelated formulas. Denoted , it is also called the electric
force constant or electrostatic constant, hence the
subscript .
The exact value of Coulomb's constant is:
The absolute value of the force
between two point
charges and relates to the distance between the point
charges and to the simple product of their charges. The
diagram shows that like charges repel each other, and
opposite charges attract each other.
Conditions for validity
When it is only of interest to know the magnitude of the
electrostatic force (and not its direction), it may be
easiest to consider a scalar version of the law. The scalar
form of Coulomb's Law relates the magnitude and sign of
the electrostatic force
acting simultaneously on two
point charges and as follows:
There are two conditions to be fulfilled for the validity of
Coulomb’s law:
1. The charges considered must be point charges.
2. They should be stationary with respect to each
other.
Scalar form
where is the separation distance and
is Coulomb's
constant. If the product
is positive, the force
between the two charges is repulsive; if the product is
negative, the force between them is attractive.
Vector form
to Newton's third law, is
.
System of discrete charges
In the image, the vector
is the force experienced by
, and the vector
is the force experienced by .
When
the forces are repulsive (as in the
image) and when
the forces are attractive
(opposite to the image). The magnitude of the forces will
always be equal.
Coulomb's law states that the electrostatic force
experienced by a charge, at position
, in the
vicinity of another charge, at position , in a vacuum
is equal to:
The law of superposition allows Coulomb's law to be
extended to include any number of point charges. The
force acting on a point charge due to a system of point
charges is simply the vector addition of the individual
forces acting alone on that point charge due to each one
of the charges. The resulting force vector is parallel to
the electric field vector at that point, with that point
charge removed.
The force on a small charge, at position , due to a
system
of
discrete
charges
in
vacuum
is:
where
Where
,
, and
the
unit
vector
is the electric constant.
The vector form of Coulomb's law is simply the scalar
definition of the law with the direction given by the unit
vector,
, parallel with the line from charge
to
charge .[14] If both charges have the same sign (like
charges) then the product
is positive and the
direction of the force on is given by
; the charges
repel each other. If the charges have opposite signs then
the product
is negative and the direction of the force
on is given by
; the charges attract each other.
The electrostatic force
experienced by
, according
and
respectively of the
direction of
charges to )
are the
charge,
magnitude and position
is a unit vector in the
(a vector pointing from
Continuous charge distribution
In this case, the principle of linear superposition is also
used. For a continuous charge distribution, an integral
over the region containing the charge is equivalent to an
infinite summation, treating each infinitesimal element of
space as a point charge
. The distribution of charge is
usually linear, surface or volumetric.
For a linear charge distribution (a good approximation
for charge in a wire) where
unit length at position
, and
element of length,
gives the charge per
is an infinitesimal
.
For a surface charge distribution (a good approximation
for charge on a plate in a parallel plate capacitor) where
gives the charge per unit area at position
is an infinitesimal element of area,
, and
Experiment to verify Coulomb's law.
For a volume charge distribution (such as charge within a
bulk metal) where
at position , and
volume,
gives the charge per unit volume
is an infinitesimal element of
The force on a small test charge at position in
vacuum is given by the integral over the distribution of
charge:
Simple experiment to verify Coulomb's law
It is possible to verify Coulomb's law with a simple
experiment. Let's consider two small spheres of mass
and same-sign charge , hanging from two ropes of
negligible mass of length . The forces acting on each
sphere are three: the weight
, the rope tension and
the electric force .
Electric Field Intensity
The concept of an electric field was introduced. It was
stated that the electric field concept arose in an effort to
explain action-at-a-distance forces. All charged objects
create an electric field that extends outward into the
space that surrounds it. The charge alters that space,
causing any other charged object that enters the space to
be affected by this field. The strength of the electric field
is dependent upon how charged the object creating the
field is and upon the distance of separation from the
charged object. In this section of Lesson 4, we will
investigate electric field from a numerical viewpoint - the
electric field strength.
If the electric field strength is denoted by the symbol E,
then the equation can be rewritten in symbolic form as
.
The Force per Charge Ratio
Electric field strength is a vector quantity; it has both
magnitude and direction. The magnitude of the electric
field strength is defined in terms of how it is measured.
Let's suppose that an
electric charge can be
denoted by the symbol
Q. This electric charge
creates an electric field;
since Q is the source of the electric field, we will refer to
it as the source charge. The strength of the source
charge's electric field could be measured by any other
charge placed somewhere in its surroundings. The charge
that is used to measure the electric field strength is
referred to as a test charge since it is used to test the
field strength. The test charge has a quantity of charge
denoted by the symbol q. When placed within the electric
field, the test charge will experience an electric force either attractive or repulsive. As is usually the case, this
force will be denoted by the symbol F. The magnitude of
the electric field is simply defined as the force per charge
on the test charge.
The standard metric units on electric field strength arise
from its definition. Since electric field is defined as a
force per charge, its units would be force units divided by
charge units. In this case, the standard metric units are
Newton/Coulomb or N/C.
In the above discussion, you will note that two charges
are mentioned - the source charge and the test charge.
Two charges would always be necessary to encounter a
force. In the electric world, it takes two to attract or repel.
The equation for electric field strength (E) has one of the
two charge quantities listed in it. Since there are two
charges involved, a student will have to be ultimately
careful to use the correct charge quantity when
computing the electric field strength. The symbol q in the
equation is the quantity of charge on the test charge (not
the source charge). Recall that the electric field strength
is defined in terms of how it is measured or tested; thus,
the test charge finds its way into the equation. Electric
field is the force per quantity of charge on the test
charge.
The electric field strength is not dependent upon the
quantity of charge on the test charge. If you think about
that statement for a little while, you might be bothered by
it. (Of course if you don't think at all - ever - nothing
really bothers you. Ignorance is bliss.) After all, the
quantity of charge on the test charge (q) is in the
equation for electric field. So how could electric field
strength not be dependent upon q if q is in the equation?
Good question. But if you think about it a little while
longer, you will be able to answer your own question.
(Ignorance might be bliss. But with a little extra thinking
you might achieve insight, a state much better than bliss.)
Increasing the quantity of charge on the test charge - say,
by a factor of 2 - would increase the denominator of the
equation by a factor of 2. But according to Coulomb's
law, more charge also means more electric force (F). In
fact, a twofold increase in q would be accompanied by a
twofold increase in F. So as the denominator in the
equation increases by a factor of two (or three or four),
the numerator increases by the same factor. These two
changes offset each other such that one can safely say
that the electric field strength is not dependent upon the
quantity of charge on the test charge. So regardless of
what test charge is used, the electric field strength at any
given location around the source charge Q will be
measured to be the same.
Coulomb's law equation. Coulomb's law states that the
electric force between two charges is directly
proportional to the product of their charges and inversely
proportional to the square of the distance between their
centers. When applied to our two charges - the source
charge (Q) and the test charge (q) - the formula for
electric force can be written as
Another Electric Field Strength Formula
Note that the derivation above shows that the test charge
q was canceled from both numerator and denominator of
the equation. The new formula for electric field strength
(shown inside the box) expresses the field strength in
terms of the two variables that affect it. The electric field
strength is dependent upon the quantity of charge on the
source charge (Q) and the distance of separation (d) from
The above discussion pertained to defining electric field
strength in terms of how it is measured. Now we will
investigate a new equation that defines electric field
strength in terms of the variables that affect the electric
field strength. To do so, we will have to revisit the
If the expression for electric force as given by Coulomb's
law is substituted for force in the above E =F/q equation,
a new equation can be derived as shown below.
the source charge.
separation distance decreases by a factor of 2, the electric
field strength increases by a factor of 4 (2^2).
An Inverse Square Law
Like all formulas in physics, the formulas for electric
field strength can be used to algebraically solve physics
word problems. And like all formulas, these electric field
strength formulas can also be used to guide our thinking
about how an alteration of one variable might (or might
not) affect another variable. One feature of this electric
field strength formula is that it illustrates an inverse
square relationship between electric field strength and
distance. The strength of an electric field as created by
source charge Q is inversely related to square of the
distance from the source. This is known as an inverse
square law.
Electric field strength is location dependent, and its
magnitude decreases as the distance from a location to
the source increases. And by whatever factor the distance
is changed, the electric field
strength
will
change
inversely by the square of
that factor. So if separation
distance increases by a
factor of 2, the electric field
strength decreases by a factor of 4 (2^2). If the separation
distance increases by a factor of 3, the electric field
strength decreases by a factor of 9 (3^2). If the separation
distance increases by a factor of 4, the electric field
strength decreases by a factor of 16 (4^2). And finally, if
Use this principle of the inverse square relationship
between electric field strength and distance to answer the
first three questions in the Check Your Understanding
section below.
The Direction of the Electric Field Vector
As mentioned earlier, electric field strength is a vector
quantity. Unlike a scalar quantity, a vector quantity is not
fully described unless there is a direction associated with
it. The magnitude of the electric field vector is calculated
as the force per charge on any given test charge located
within the electric field. The force on the test charge
could be directed either towards the source charge or
directly away from it. The precise direction of the force
is dependent upon whether the test charge and the source
charge have the same type of charge (in which repulsion
occurs) or the opposite type of charge (in which
attraction occurs). To resolve the dilemma of whether the
electric field vector is directed towards or away from the
source charge, a convention has been established. The
worldwide convention that is used by scientists is to
define the direction of the electric field vector as the
direction that a positive test charge is pushed or pulled
when in the presence of the electric field. By using the
convention of a positive test charge, everyone can agree
upon the direction of E.
LECTURE NO.15
Electrostatics : Electric Field & Potential
Electrostatic Potential :
Consider the expression for the electric field for a
continuous charge distribution
Thus we have two following relations
LECTURE NO.17
Electric field due to Continuous charge Distributions
The gradient operator can be taken outside as the
gradient is with respect to unprimed variable , while the
integration is with respect to primed variables which
gives
Point Charge: Charge whose volume is very small when
compared to the distance of separation under consideration.
Line Charge: Charge whose surface area is very small when
compared to its length.
Surface Charge: Charge whose thickness is very small when
compared to its area.
The divergence of the Electric field can be written as
The Laplacian operator acting on unprimed variable can
be taken inside the integration
Volume Charge: A charged object comprising of three
dimensions. Coulomb’s law can be applied to find the force
exerted on some point charge q due to line, surface & volume
distribution of charges. Here the charges are not
concentrated rather they are distributed. In this situation the
total sum must be replaced by integral since a continuous
distribution of charges are taken care of.
Electric field due to Continuous Uniform charge
Distributions
The charge density (  L in C/m), surface charge density (  S in
C/m2), volume charge density (  V in C/m3)
such as a very fine, sharp beam in a cathode ray tube of very
small radius it is convenient to heat the charge as a line
charge density. its unit is Coulombs/m.
Surface charge density (  S )
Line charge dQ   L dl
One basic charge configuration often used to approximate
that found on the conductors of a strip transmission line or
parallel plate capacitor is the infinite sheet of charge having a
Surface charge dQ   S ds
reside on the surface of conductor & not in their interiors,
uniform charge density of  S Coulomb/m2.The static charges
hence  S is known as surface charge density.
Volume charge dQ   v dv
  S dS a (Surface
Hence Electric field intensity E 
R
4 0 R 2
Electric field due to Continuous charge Distribution due to
Line charge:
charge)

E
dV 
a R (Volume charge)
4 0 R 2
V
Volume Charge density (  V )
The space between the control grid and thee cathode in the
electron gun assembly of a cathode ray tube operating with
space charge, we can replace this distribution of very small
particles with a smooth continuous distribution described by
a volume charge density and its unit is Coulombs/m3.
Line charge density (  L )
If a filament like distribution of volume charge is considered
wo point charges. Our variation tells us the Electric field
due to a single point charge. What do we do if we have a
continuous charge distribution? We can sum up the
electric field caused by each tiny, infinitesimal part of the
charge distribution. This means an integral over the
charge distribution:
For a single point charge Q, we had
where r is the distance from the charge Q. Remember, E
is only the magnitude of the electric field; we must take
care of its vector nature separately! That's important!
Now we have a distribution of charge and we must
replace Q by dQ and E by dE -- and take care of the
direction of E.
r, the distance from the tiny, elemental, infinitesimal
charge dQ to the point in question, is a function of where
that charge dQ is. And, what does it mean to "integrate
over the charge dQ"? We know how to integrate over a
variable like dx, or a plane like dA = dx dy or dA = 2 r
dr or dA = r d dr, or a volume like dV = dx dy dz. So we
will need to change from this symbolic charge dQ to a
charge density multiplied by some spatial differential,
What is the electric field at point P because of a little
piece of charge Q located at position x, as shown in the
sketch?
E = k Q / x2
We will carry out an integration from x = d to x = d + so
we need to change this small amount of charge Q into a
small length x,
Q=
dQ = dx
E=k(
x
x) / x2
dQ = dA
where
dQ = dV
=Q/
Look at Example in Serway's and Beichner's textbook
(p.724): A rod of length has a uniform charge per unit
length and a total charge Q. Calculate the electric field
at a point P along the axis of the rod, a distance d from
one end.
dE = k ( dx) / x2
dE = k (dx / x2)
The charge density is
= Q / (2 a)
Remember, our equation for the electric field is for the
magnitude of the electric field. Consider a little piece of
charge dq as sketched in the diagram. Because that
charge dq is there, there is an electric field dE at point P
in the direction shown. The component dEx of that
electric field along the direction of the axis perpendicular
to the plane of the ring is
dEx = dE cos
dEx = dE (x/r)
What about other geometries?
dEx = [k dq/r2] (x/r)
dEx = [k dq/r3] x
Find the electric field due to a ring of charge: A ring of
radius a has a uniform charge density with a total charge
Q. Calculate the electric field along the axis of the ring at
a point P, a distance x from the center of the ring.
dEx = [k x dq/r3]
dEx = [k x/r3] dq
Notice that, with this geometry, once the radius of the
ring a is specified and the position x, that fully specifies
r. r and x do not change as we integrate over dq.
important. Don't start writing equations before you have
made good, clear, complete diagrams!
LECTURE NO.18
r = SQRT(a2 + x2)
r3 = (a2 + x2)3/2
Electric Flux Density
The Electric Flux Density (D) is related to the Electric Field (E) by:
1/r3 = 1/(a2 + x2)3/2
In the aboveequation ,
is the permittivity of the medium (material) where we a
If you recall that the Electric Field is equal to the force per unit charge
Then the Electric Flux Density is:
Remember, x and a are not variables.
What about the component of E that is perpendicular to
this direction? By symmetry that component is zero.
From the diagram, you can see that for each element of
charge dq, there is another element of charge dq on the
opposite side of the ring that causes an electric field that
just cancels the first one -- that is, their components
perpendicular to the axis of symmetry just cancel. Notice
that their components along the axis do not cancel for
they lie in the same direction. Diagrams are very
From the above equation the Electric Flux Density is very similar to the Electric F
the material in which we are measuring (that is, it does not depend on the permitti
field is a vector field, which means that at every point in space it has a magnitude
The Electric Flux Density has units of Coulombs per meter squared [C/m^2].
LECTURE NO.19 Gauss's law
For Gauss's theorem, a mathematical theorem relevant to
all of these laws, Divergence theorem. In physics,
Gauss's law, also known as Gauss's flux theorem, is a law
relating the distribution of electric charge to the resulting
electric field.
Qualitative description of the law
In words, Gauss's law states that:
Equation involving E field
Gauss's law can be stated using either the electric field E
or the electric displacement field D. This section shows
some of the forms with E; the form with D is below, as
are other forms with E.
Integral form
Gauss's law may be expressed as:
The net electric flux through any closed surface is
equal to 1⁄ε times the net electric charge enclosed
within that closed surface.
Gauss's law has a close mathematical similarity with a
number of laws in other areas of physics, such as Gauss's
law for magnetism and Gauss's law for gravity. In fact,
any "inverse-square law" can be formulated in a way
similar to Gauss's law: For example, Gauss's law itself is
essentially equivalent to the inverse-square Coulomb's
law, and Gauss's law for gravity is essentially equivalent
to the inverse-square Newton's law of gravity.
Gauss's law is something of an electrical analogue of
Ampère's law, which deals with magnetism.
The law can be expressed mathematically using vector
calculus in integral form and differential form, both are
equivalent since they are related by the divergence
theorem, also called Gauss's theorem. Each of these
forms in turn can also be expressed two ways: In terms of
a relation between the electric field E and the total
electric charge, or in terms of the electric displacement
field D and the free electric charge.
where ΦE is the electric flux through a closed surface S
enclosing any volume V, Q is the total charge enclosed
within S, and ε0 is the electric constant. The electric flux
ΦE is defined as a surface integral of the electric field:
where E is the electric field, dA is a vector representing
an infinitesimal element of area, and · represents the dot
product of two vectors.
Since the flux is defined as an integral of the electric
field, this expression of Gauss's law is called the integral
form.
Applying the integral form
If the electric field is known everywhere, Gauss's law
makes it quite easy, in principle, to find the distribution
of electric charge: The charge in any given region can be
deduced by integrating the electric field to find the flux.
However, much more often, it is the reverse problem that
needs to be solved: The electric charge distribution is
known, and the electric field needs to be computed. This
is much more difficult, since if you know the total flux
through a given surface that gives almost no information
about the electric field, which (for all you know) could
go in and out of the surface in arbitrarily complicated
patterns.
An exception is if there is some symmetry in the
situation, which mandates that the electric field passes
through the surface in a uniform way. Then, if the total
flux is known, the field itself can be deduced at every
point. Common examples of symmetries which lend
themselves to Gauss's law include cylindrical symmetry,
planar symmetry, and spherical symmetry.
equivalent, by the divergence theorem. Here is the
argument more specifically.
The integral form of Gauss's law is:
for any closed surface S containing charge
Q. By the divergence theorem, this equation
is equivalent to:
for any volume V containing charge Q. By
the relation between charge and charge
density, this equation is equivalent to:
Differential form
By the divergence theorem Gauss's law can alternatively
be written in the differential form:
where ∇ · E is the divergence of the electric field, ε0 is
the electric constant, and ρ is the total electric charge
density.
Equivalence of integral and differential forms
The integral and differential forms are mathematically
for any volume V. In order for this equation
to be simultaneously true for every possible
volume V, it is necessary (and sufficient) for
the integrands to be equal everywhere.
Therefore, this equation is equivalent to:
Thus the integral and differential forms are
equivalent.
Equation involving D field
where ΦD is the D-field flux through a surface S which
encloses a volume V, and Qfree is the free charge
contained in V. The flux ΦD is defined analogously to the
flux ΦE of the electric field E through S:
Free, bound, and total charge
The electric charge that arises in the simplest textbook
situations would be classified as "free charge"—for
example, the charge which is transferred in static
electricity, or the charge on a capacitor plate. In contrast,
"bound charge" arises only in the context of dielectric
(polarizable) materials. (All materials are polarizable to
some extent.) When such materials are placed in an
external electric field, the electrons remain bound to their
respective atoms, but shift a microscopic distance in
response to the field, so that they're more on one side of
the atom than the other. All these microscopic
displacements add up to give a macroscopic net charge
distribution, and this constitutes the "bound charge".
Although microscopically, all charge is fundamentally
the same, there are often practical reasons for wanting to
treat bound charge differently from free charge. The
result is that the more "fundamental" Gauss's law, in
terms of E (above), is sometimes put into the equivalent
form below, which is in terms of D and the free charge
only.
Integral form
Differential form
The differential form of Gauss's law, involving free
charge only, states:
where ∇ · D is the divergence of the electric
displacement field, and ρfree is the free electric charge
density.
Equation for linear materials
In homogeneous, isotropic, nondispersive, linear
materials, there is a simple relationship between E and D:
where ε is the permittivity of the material. For the case of
vacuum (aka free space), ε = ε0. Under these
circumstances, Gauss's law modifies to
This formulation of Gauss's law states the total charge
form:
for the integral form, and
with respect to r, and use the known theorem
for the differential form.
Gauss's law can be derived from Coulomb's law.
Coulomb's law states that the electric field
due to a stationary point charge is:
where δ(r) is the Dirac delta function, the
result is
Using the "sifting property" of the Dirac
delta function, we arrive at
where
er is the radial unit vector,
r is the radius, |r|,
is the electric constant,
q is the charge of the particle, which
is assumed to be located at the origin.
Using the expression from Coulomb's law,
we get the total field at r by using an integral
to sum the field at r due to the infinitesimal
charge at each other point s in space, to give
where is the charge density. If we take the
divergence of both sides of this equation
which is the differential form of Gauss's law,
as desired.
Deriving Coulomb's law from Gauss's law
Strictly speaking, Coulomb's law cannot be derived from
Gauss's law alone, since Gauss's law does not give any
information regarding the curl of E (see Helmholtz
decomposition and Faraday's law). However, Coulomb's
law can be proven from Gauss's law if it is assumed, in
addition, that the electric field from a point charge is
spherically-symmetric (this assumption, like Coulomb's
law itself, is exactly true if the charge is stationary, and
approximately true if the charge is in motion).
charge of magnitude q as shown below in the
figure.
Taking S in the integral form of Gauss's law
to be a spherical surface of radius r, centered
at the point charge Q, we have
By the assumption of spherical symmetry,
the integrand is a constant which can be
taken out of the integral. The result is
where is a unit vector pointing radially
away from the charge. Again by spherical
symmetry, E points in the radial direction,
and so we get

which is essentially equivalent to Coulomb's
law. Thus the inverse-square law
dependence of the electric field in Coulomb's
law follows from Gauss's law.
Application of gauss law
(A) Derivation of Coulumb's Law


Coulumb's law can be derived from Gauss's law.
Consider electric field of a single isolated positive


Field of a positive charge is in radially outward
direction everywhere and magnitude of electric
field intensity is same for all points at a distance r
from the charge.
We can assume Gaussian surface to be a sphere
of radius r enclosing the charge q.
From Gauss's law
since E is constant at all points on the surface
therefore,

We can assume Gaussian surface to be a right
circular cylinder of radius r and length l with its
ends perpandicular to the wire as shown below in
the figure.

λ is the charge per unit length on the wire.
Direction of E is perpandicular to the wire and
components of E normal to end faces of cylinder
makes no contribution to electric flux. Thus from
Gauss's law

Now consider left hand side of Gauss's law
surface area of the sphere is A=4πr2
thus,

Now force acting on point charge q' at distance r
from point charge q is
This is nothing but the mathematical statement of
Coulumb's law.
(B) Electric field due to line charge



Consider a long thin uniformly charged wire and
we have to find the electric field intensity due to
the wire at any point at perpandicular distance
from the wire.
If the wire is very long and we are at point far
away from both its ends then field lines outside
the wire are radial and would lie on a plane
perpandicular to the wire.
Electric field intensity have same magnitude at all
points which are at same distance from the line
charge.
Since at all points on the curved surface E is
constant. Surface area of cylinder of radius r and
length l is A=2πrl therefore,

Charge enclosed in cylinder is q=linear charge
density x length l of cylinder,
or, q=λl
From Gauss's law
Thus electric field intensity of a long positively
charged wire does not depends on length of the
wire but on the radial distance r of points from the
wire.
LECTURE NO.19
.