Download PH504lec0809-2

PH504
Part 2. Electrostatics
Coulomb’s law and electric field (E-field)
1. Introduction
Electromagnetism is concerned with the study of the
properties of charge, one of the fundamental properties of
nature.
 There are two types of charge; positive and negative.
Potatoes don’t explode; in bulk matter, their effects are
almost completely neutralized.
 Charge can neither be created nor destroyed. However
positive and negative charges act to cancel each other.
E.g.
238
92
4
U234
90Th  2 He
:92 protons on each side.
 Charge is quantized: +e, -e (free quarks with 2/3 and
1/3 these values don’t exist in nature).The unit of charge is
the Coulomb (C) the usual mathematical symbol is Q :
e = 1.602 ´ 10-19 C
The movement of charge constitutes an electric current.
If a charge dQ passes a given point in a time dt then a
current I=dQ/dt is said to occur.
Method of inducing charge:
2. Definitions
(i) The point charge. Assumes all the charge is
concentrated at a point having zero volume. If charge is
spread over a finite volume then it approximates to a point
charge if its physical extent is small compared to the
distance(s) to other charges.
(ii) Volume charge density:  Charge is spread over a
finite volume. Density at a given point is r (units C m-3). r
is not necessary constant. The total charge Q contained
within a volume  is given by
is performed over the volume .
where the integral
(iii) Surface charge density: . Charge is spread thinly
over a surface or a sheet. The density at a given point is 
(units C m-2). The total charge Q on a whole surface S is
given by:
the surface S.
where the integral is performed over
(iv) Line charge density: . Charge is distributed along a
line. The density at a given point is  (units C m-1). The
total charge Q in a total length L is given by
where the integral is performed along the line
L.
Electrostatics. Concerned with the properties of charges
which are stationary. Although we will need to move
charges when deriving equations for potential energy etc,
the charges can always be taken to move infinitesimally
slowly.
For Electrostatics:
 is the charge density
J = 0 is the current density
3. Forces between stationary charges (in vacuo) –
Coulomb’s law - Charles Auguste Coulomb - 1785
Experiments show that an electric force exists between
two charges.
The size of this force is proportional to the product of the
magnitudes of the two charges.
It is inversely proportional to the square of their
separation.
The force acts along the line joining the charges.
It is repulsive for charges of identical (like) sign and
attractive for opposite sign charges.
• Like charges repel
• Unlike charges attract:
like charges
unlike charges
For two point charges Q1 and Q2 separated by a distance r
in a vacuum, the electric force is described by Coulomb’s
law:
F = Q1Q2/(40r2)
(scalar form)
in magnitude, along the r direction.
F = Q1Q2
where
/(40r2)
is a unit vector along r, or
(vector form)
F = Q1Q2 r/(40r3)
(vector form)
 is a constant which gives the strength of the electric
force.  is known as the permittivity of free space. 

= 8.8542x10-12 C2 m-2 N-1 or F m-1.
E = 1.6 x 10-19 C
Principle of superposition
For a system consisting of three or more charges, the
electric force acting on any one charge is given by the
vector summation of the individual forces due to all the
other charges.
4.The electric field (E-field)
Frequently we wish to investigate the force (and
subsequent motion) on an arbitrary charge due to a set of
other known fixed charges. Although Coulomb’s law can
be used it is generally more convenient to think of the
fixed charges as producing a field, the electric or E-field,
which then exerts a force on any charge placed in the
field.
What is a ‘field’? An electric field is defined in a region of
space where another charge would be influenced by a
charge or distribution of charges.
If a test charge Qt, placed at some position in space,
experiences an electric force F then the E-field at that
point is given by
where the limit Qt0 is required so that Qt does not
perturb the charges which produce F and E.
The units of E are N C-1 or more usual V m-1.
E-field due to a point charge
For a single point charge Q
the field points radially outwards for a positive charge and
radially inwards for a negative charge.
For a collection of two or more point charges the principle
of superposition can be applied to find the total E-field at a
given point.
For continuous charge distributions the distribution is split
up into an infinite number of infinitesimally small,
equivalent point charges with the E-field then being given
by a suitable integration.
The electric flux through a closed surface is proportional to
the charge enclosed.
Flux: The amount of field, material or other physical entity
passing through a surface. Surface area can be
represented as vector defined normal to the surface
The flux of E through the surface of a sphere can be
written as Gauss’s law in integral form:
 Ed A 
S
Q
o
since
 
1 q
E r  
rˆ
4o r 2
For a sphere this is obvious:
This holds for any shaped surface!
In terms of the divergence, using the divergence theorem:
div 0E = (Gauss’s Law)

Techniques for finding the E-field will be developed later in
this course.
Secondly, since the field is purely divergent (each point
charge generates a radial field dependent only on r):
curl E = 0 .
Solving Problems with Gauss’ Law
Choose a coordinate system that most nearly matches the
symmetry of the charge distribution. For example, we
chose spherical coordinates to determine the flux due to a
point charge because of spherical symmetry.
Example: Co-axial cable.
Consider a cylindrical surface of radius R.
Take a charge per unit length 
Choose the following Gaussian surface:
Then Gauss's Law yields
E 2r = o E = 2ro) for r > R.
E=0
for r < R
………………..much easier than integrating.
Electric Field Lines
These allow the form of the E-field to be visualised in a
limited sense: dx/Ex = dy/Ey = dz/Ez
The lines have the following properties.
 The tangent to the lines at any point gives the direction
of the E-field at that point.
 Lines start on positive charges and finish on negative
ones (but these are not particle/charge trajectories).
 The density of lines gives an indication of the field
strength at a given point.
Equal charges:
Conclusions

Charge and relationship to current

Definitions of point, volume, surface and line charges

Coulomb’s law for two point charges

Superposition and forces between >2 point charges

Definition of E-field

E-fields resulting from one or more point charges

E-fields due to continuous charge distributions

E-field line diagrams
THE END