Download 1.2. Polar Form

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PH504: 1. Mathematics
The mathematics required for PH504.
Electromagnetism requires a rigorous mathematical
treatment.
Objective: to provide a revision of the most important and
relevant points.
This is not intended to teach the
mathematics from scratch. For more detailed treatments
consult your first year mathematics’ notes or textbooks.
Some of the electromagnetism textbooks provide a
chapter or appendix covering the required mathematics.
Examples from lectures to come:
XE  
B
t
H  H 0 exp[ j(K  r  t )]
E  E 0 exp[ j(K  r  t )]
and
XH  X{exp[ j(K  r  t )]H 0 }  jKXH 0 exp[ j(K  r  t )]  jKXH
 H  dl 

 X H  dS 
Sinside
 J  dS  0
Sinside
Can you cope???
1. Complex Numbers.
Positive Scalars. The voltage produced by a battery is a
scalar quantity. So is the resistance of a piece of wire
(ohms), or the current through it (amps).
Negative numbers: debt
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Irrational numbers: We need square roots to evaluate the
diagonal length of a square.
The square root of a negative number?
IMAGINERY or Impossible Number
However, when we begin to analyze alternating current
circuits, we find that quantities of voltage, current, and
even resistance (called impedance in AC) are not the
familiar one-dimensional quantities we are used to
measuring in DC circuits. Rather, these quantities,
because they're dynamic (alternating in direction and
amplitude), possess other dimensions that must be taken
into account.
Frequency and phase shift are two of these dimensions
that come into play. Even with relatively simple AC
circuits, where we are only dealing with a single
frequency, we still have the dimension of phase shift to
contend with in addition to the amplitude.
Complex numbers. Here is where we need to abandon
scalar numbers for something better suited: complex
numbers. Just like the example of giving directions from
one city to another, AC quantities in a single-frequency
circuit have both amplitude (analogy: distance) and phase
shift (analogy: direction).
A complex number is a single mathematical quantity able
to express these two dimensions of amplitude and phase
shift at once
A single complex variable can be used to represent two
real independent variables with the same dimensions.
E.g. Coordinates, velocities, streamfunction/potential
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Vectors. .When used to describe an AC quantity, the
length of a vector or phasor represents the amplitude of
the wave while the angle of a vector represents the phase
angle of the wave relative to some other (reference)
waveform.
Complex numbers are useful for AC circuit analysis
because they provide a convenient method of symbolically
denoting phase shift between AC quantities like
voltage and current.
i or j = square root of –1. A complex number consists of
a real and an imaginary part. Both the real and imaginary
parts are real numbers, but the imaginary part is multiplied
with the square root of -1. Complex numbers can be
expressed in numerous forms.
One REAL elephant:
One IMAGINERY ghost:
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Complex numbers are expressed in three forms………
1.1 Rectangular Form
A complex number in rectangular form looks like this

A = a + jb
where a is the real part and b is the imaginary part. j is sqrt(-1).
The concept is: Imaginery Numbers occupy an axis
perpendicular to that occupied by Real Numbers.
Adding and subtracting complex numbers in rectangular form is
carried out by adding or subtracting the real parts and then adding
and subtracting the imaginary parts.


(5 + j2) + (2 - j7) = (5 + 2) + j(2 - 7) = 7 - j5
(2 + j4) - (5 + j2) = (2 - 5) + j(4 - 2) = -3 + j2
Multiplying is slightly harder than addition or subtraction. It must
be carried out like the multiplication of two binomials, multiplying
both parts of one by both parts of the other.
When you multiply the imaginary parts,
j*j = sqrt(-1) * sqrt(-1) = -1, so that part becomes real).

(2 + j2) (8 - j3) = (2 * 8) + j(2 * 8) + j(2 * -3) + j*j (2 * -3) = 16
+ j16 - j6 + 6 = 22 + j10
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Concept: multiplying by j rotates the number by a right angle, 90
degrees.
Division requires a new idea to be introduced. The complex
conjugate of a number is the number that has the same real part
as the original number but an imaginary part that differs only in its
sign. The complex conjugate is denoted by an asterisk
immediately following the number or variable.


A = (2 + j2)
A* = (2 + j2)* = 2 - j2
When dividing two complex numbers, you must first multiply both
the numerator and denominator by the complex conjugate of the
denominator. This multiplication results in a denominator that has
only a real part.

(4 + j3) / (2 + j2) = ((4 + j3) (2 - j2)) / ((2 + j2) (2 - j2)) = ((8 +
6) + j(8 - 6)) / ((4 + 4) + j(4 - 4)) = (14 + j2) / 8
= 7/4 + j/4
But how would you deal with (1+ j)8
??
1.2. Polar Form
Another way to represent complex numbers is in polar form. If you
look at the real and imaginary parts of a complex number as
coordinates in a plane, then the real part would be the x coordinate
and the imaginary part the y coordinate. In rectangular form, the x
and y coordinate are specified in that way.
In polar form, the point in the plane is instead defined by a
magnitude and an angle. Polar form relates to rectangular form in
the following way.
(magnitude) r = √ (a² + b²)



argument(angle) theta = tan-1 b/a
and
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

a = r cos (theta)
b = r sin (theta) .
A complex number is then represented as

A = r | theta
where
r = magnitude = |A| and
theta = angle = ang A
Multiplication and division are much simpler for numbers in
polar form. Rotation is easy.
To find the conjugate of a complex number in polar form, simply
reverse the sign of the angle.


A = 5 | 25°
A* = 5 | -25°
With multiplication you multiply the magnitudes and add the
angles, and with division you divide the second magnitude from
the first and subtract the second angle from the first.


(5 | 45° ) (2 | 20°) = (5)(2) | (45 + 20)° = 10 | 65°
(4 | 90°) / (2 | 45°) = 4/2 | (90 - 45)°
= 2 | 45°
To add or subtract we basically need to convert the
numbers back into rectangular form

(2 | 45°) + (8 | 30°)
= (2 cos 45° + 8 cos 30°) + j(2 sin 45° + 8 sin 30°)
= 8.342 + j2.414
The answer can then be converted back to polar form if
desired


r = √ (8.342² + 2.414²) = 8.684
theta = tan-1 (2.414/8.342) = 16.1°
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
(2 | 45°) + (8 | 30°) = 8.684 | 16.1°
If possible it is best to add and subtract in rectangular
form and multiply and divide in rectangular form.
(1+ j)8
= …………..
1.3. Complex Exponential Form
The complex exponential form is another way of representing a
complex number. The following formula shows how it relates to
rectangular and polar form
e j = cos () + j sin () = 1 | 
Why this form? Because it satisfies several conditions and
definitions.Complex numbers are often used to represent waves,
and wavefunctions.
The complex exponential works in the same way as polar form,
multiplication and division are carried out simply by multiplying (for
multiplication) or dividing (for division) the coefficients, and then
adding (for multiplication) or subtracting (for division) the angle.
The complex exponential is important for deriving formulas and is
the basis for some of the methods of circuit analysis, but from an
algebraic standpoint it behaves in a similar way as polar form.
Example 1: Express a sinusoidal time variation with frequency nu
and amplitude A in complex exponential form. What is the angular
frequency? Differentiate with respect to time.
Example 2: the exponential, imaginary and irrational pi are three
special numbers in mathematics.
Evaluate
e i .
Example 3: what is the square root of 5 – 12j ?
Importance for waves : Fourier analysis (and wave can be
split up into a series of sines and cosines) and AC Circuits.
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See:
http://farside.ph.utexas.edu/teaching/315/Waves/node72.html
2. Partial Differentiation
Many physical quantities are a function of more than one
variable (e.g. the pressure of a gas depends upon both
temperature and volume, a magnetic field may be a
function of the three spatial co-ordinates (x,y,z) and time
(t)). Hence when differentiating a function there is usually
a choice of which variable we differentiate with respect to.
For example consider the function f which depends upon
the variables x and y (f(x,y)).
We can differentiate f with respect to x or y. When we
differentiate with respect to a given variable we proceed in
the same manner as in basic differentiation for functions
which depend upon one variable.
However for functions of more than one variable all other
variables are treated as if they are constants. The symbols
for the differential are modified (‘’ is used instead of ‘d’).
The derivatives of f with respect to x or y are written as
respectively.
Example: Why is df/dx = 2x when f = x2?
Why is df/dx = f when f = e x = exp(x) ?
Example: f(x,y) = 3x3+yx2+4xy2+5y3
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We can also take higher order derivatives, e.g.
differentiate twice with respect to x
or differentiate with respect to one variable (say x ) and
then a second (y)
Note that for all well behaved functions the order in which
we differentiate (e.g. x then y or y then x) is unimportant,
i.e.
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3. Angles and Solid Angles
Angles. We need to consider both normal (onedimensional) and solid (two-dimensional angles)
In (a) the line of length dl has a component dl cosalong
the arc of the circle. The subtended angle is defined as
this component divided by the radius of the circle r.
Radians. The units are radians. As the circumference of a
circle is 2r there are 2r/r=2 radians in a circle.
In (b) the area da makes an angle  to the surface of the
sphere of radius r. The projection of da onto the surface of
the sphere is hence da cos  and the solid angle is
defined as this projection divided by the square of the
radius
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The units are steradians. As the area of a sphere is 4r2
there are 4r2/r2=4 steradians in a sphere.
4. Vectors – Vector Analysis
Many physical quantities have a direction as well as a
magnitude. Examples are force, velocity and, in
electromagnetism, electric and magnetic fields. We
describe such quantities using vectors.
At each point in space we can imagine an arrow whose
length gives the magnitude of the quantity it describes and
whose direction corresponds to the direction of the
quantity.
We want: displacement + angular displacement
Components. In dealing with vectors it is often convenient
to describe a vector in terms of components. Because
space has three-dimensions, three components lying
along three orthogonal directions are required to describe
any vector.
The most common system is the Cartesian one where
the three directions are the x, y and z-axes.
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Unit Vectors. To define any vector A in the Cartesian
system we need the size of the components along the
three axes (Ax, Ay and Az) and three unit vectors that are
parallel to the three axes (i, j and k parallel to x, y and z
respectively).
The vector A is given by
The magnitude of A is given by
4.1 Non-Cartesian Systems
The Cartesian system is the most common one and the
easiest to visualise and use. Two other systems are useful
when considering problems with
cylindrical (wires or cables) and
spherical symmetry (planets).
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In the above cylindrical system the three components of
the vector are defined as lying along the radial direction in
the x-y plane (r), the angle between the projection onto the
x-y plane and the x-axis () and the vertical or z
component (z).
In the spherical system the three components are along
the radial direction (r), the angle between the projection
onto the x-y plane and the x-axis () and the angle
between the vector and the z-axis ().
4.2 Multiplication of Vectors
Addition of Vectors: associative and commutative
There are two types of multiplication, the dot product,
resulting in a scalar and the vector product resulting in a
vector.
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The dot product of two vectors (scalar product)
If we have two vectors A and B then the dot product of A
and B is defined as
where A and B are the magnitudes of vectors A and B
and  is the angle between the two vectors.
The dot product is commutative
Physically the dot product represents the projection of
one of the vectors on to the other times the magnitude of
the other. If the two vectors are mutually perpendicular
then the dot product is zero (cos 90o=0).
In Cartesian co-ordinates, if we have two vectors A and
B with components (Ax, Ay, Az) and (Bx, By, Bz)
respectively then
Examples:
i.i = j.j = k.k = 1
i.j = j.i = i.k = k.i = j.k = k.j = 0.
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Physical application of the dot product
We know from mechanics that if a force F moves through
a distance L then the work done is equal to the component
of the force along the direction of movement multiplied by
L.
In the diagram, the component of F along the direction
of movement is F cos. Hence the work done is FLcos .
However if we use vectors F and L to describe the
force and the movement respectively we have the
definition of the dot product. Hence when a force F is
moved by a distance L the work done is simply the dot
product of the two vectors.
This is an application of the dot product which we will use
many times in the electromagnetism course.
The cross product of two vectors (vector
product)
If we have two vectors A and B then the cross product of
A and B is defined as
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AB = AB sin n
where n is a unit vector normal to the plane containing
the two vectors A and B and whose direction is given by
the right-hand rule.
The cross product of two vectors is not commutative as
sin(-) = - sin.
AB = - BA
In Cartesian co-ordinates the cross product of the
vectors A and B with components (Ax, Ay, Az) and (Bx,
By, Bz) is given by
AB = (AyBz-AzBy)i + (AzBx-AxBz)j + (AxBy-AyBx)k
The cross product of a vector with itself is zero as the
angle between the two vectors is 0 and sin(0)=0.
i x i = j x j = k x k = O, (the zero vector).
Also,
i x j = k and j x k = i and k x i = j
while j x i = - k and k x j = - i and i x k = - j.
Physical application of the cross product
A force F acts at a distance r from a point of rotation. The
torque (T) about this point is the distance from where F
acts to the point of rotation (r) multiplied by the normal
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component of F.
T = rFsin ,
where  is the angle between F and the line drawn
through the point of rotation. However if we define torque
in terms of a vector whose magnitude gives the size of the
torque and whose direction points along the axis of
rotation then
T = rF
where r is the vector from the point of rotation to the point
where F acts. The direction of T gives the sense of
rotation from the right-hand screw rule.
Triple scalar and vector products
>>>>>>>>>>>>>>>>>
scalar triple product: (a x b) . c - the answer will be a
number, not a vector. It is the volume of the parallelepiped
defined by vectors
Vector triple product: it can be shown that
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A x (B x C)=(A.C)B - (A.B)C
and
(A x B) x C = (C.A)B - (C.B)A
These are clearly vectors.
5. Calculus of scalars and vectors
Much of physics is concerned with how one quantity
varies when one or more other quantities change. As in
other areas of physics, in electromagnetism we will be
concerned with the spatial variation (or derivative) of
both scalar and vector quantities.
The mathematics can be summarised by the use of a
differential operator called 'del' (symbol ) which itself has
directional properties.
There are three physically meaningful ways in which 
can be applied to scalars and vectors:
1. it can be applied to a scalar to give a vector (gradient),
2. it can form the dot product with a vector to give a scalar
(divergence) and
3. it can form the cross product with a vector to give
another vector (curl).
Gradient 
The gradient of a scalar function f is written f or grad f
and is given in the Cartesian system by
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The resultant quantity is a vector.
e.g. if f(x,y,z)=2x2+y3+z2xy then
f= i(4x+z2y) + j(3y2+z2x) + k 2xyz
Physical significance of the gradient. At any point the
gradient of a function points in the direction
corresponding to that for which the function varies
most rapidly.
The magnitude of the gradient vector gives the size of this
maximum variation: the maximum directional derivative.
Example: If f(x,y) gives the height (or alternatively the z
co-ordinate) of a surface as a function of the x and y coordinates then at any point  f will point in the direction of
maximum slope of the surface.
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A small ball placed on the surface will tend to roll along
the direction opposite to f. The gravitational force acting
on the ball is - mgf where m is its mass and g is the
acceleration due to gravity.
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The negative sign arises because the force acts in the
opposite direction to  f.
Alternatively if we define U(x,y) as the gravitational
potential energy of the ball (U(x,y)=mg f(x,y)) then the
gravitational force = -U.
This is a general result: Force = -gradient(potential
energy). The potential energy may be gravitational,
electrical etc.
Divergence 
The divergence of a vector A is written as A or div A
and is given by
The resultant quantity is a scalar
e.g. if A = 3x2yz i + x2z2 j + z2k then .A=6xyz+2z
Sources and sinks. Physical significance. When the
divergence of a vector is positive at a given point then
there is a source of the vector field at that point. A
negative divergence implies a sink for the vector field.
We can hence think of the divergence of a vector as
telling us how much of the vector field starts (or
terminates) at a given point.
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In (a) the vector has a constant magnitude so its
divergence is zero.
In (b) the x-component increases along the x-direction.
This vector hence has a non-zero, positive divergence.
Curl 
The curl of a vector A is written as A or curl A and is
given by
=
The latter is called the determinant.
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The resultant is a vector
eg A = yzi-2x2yzj+3x2y2zk
A = (6x2yz--2x2z)i+(y-6xy2z)j+(-4xyz-z)k
Physical significance. A non-zero curl implies that the
corresponding vector field has a sort of rotational
property.
One way to look for a curl is to imagine that the vector
field corresponds to the flow of water. If we place a small
paddle wheel in the field then the presence of a non-zero
curl suggests that the wheel will rotate.
In the above examples for (a) although the field increases
along the direction in which it points it produces no
rotation of the wheel.
However in (b) the field points along x but increases
along the y-axis and hence produces a rotation of the
wheel. Hence the curl is related to how the field changes
as we move across the field. This can also be seen
because the expression for curl contains terms Ax/y etc.
To some extent curl and div are complementary. The latter
requires that the field increases when moving along the
field direction, the former that the field increases when
moving across the field direction.
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Relationships
From the definitions of grad, div and curl the following
relationships can be established
(f)=0 the curl of a gradient is equal to zero
(A)=0 the divergence of a curl is equal to zero
The Laplacian is important:
(f)=2f=
this is the divergence of a gradient – diffusion eg
conductivity and heat transfer if f is the temperature.
Note:
Question: you take the gradient of a ………field to get a
……. field
You take the divergence of a ……. Field to get a ………
field.
Non-cartesian co-ordinates
All of the previous examples are for Cartesian coordinates. For other systems related, but different,
expressions exist for grad, div and curl
e.g. in cylindrical co-ordinates the gradient is given by
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For this course you do not need to remember the
expressions for non-Cartesian systems but you need to
know how to apply them where necessary.
6. Integration
There are two main types of integration for vectors, line
and surface
Line integrals
The line integral of a vector A between the points a and b
is given by
As we move along a path between the points a and b, at
each step we take the component of A which lies along
the direction we are moving (given by the vector dl) and
multiply it by the distance we move through.
The line integral is the sum of all these individual
values as we move from a to b.
Conservative. In general the path taken between the
points a and b must be specified. However for a certain
class of vectors the result of the integral is independent of
the path taken. Such vectors are said to be conservative.
F is a conservative vector field if there is a function f such
that F = grad(f). The function f is called a potential
function for the vector field.
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If the line integral is performed around a closed path
(initial and final points are the same) then a circle is
placed on the integral symbol
If the vector A represents a force then the line integral of
A between two points gives the work done in moving
between these two points.
Surface integrals
The surface integral of the vector A over the surface S is
defined as
The surface is split into an infinite number of infinitesimally
small sections. For each section the product of the area of
the section (dS) and the component of A normal to the
surface is formed.
The integral is the sum of all these products. If the
surface is a closed one (no edges) then a circle is placed
on the integral sign
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7. Relationships between integrals
Divergence theorem (Gauss)
One of the most important results in vector field theory is
the so-called divergence theorem or Gauss' theorem. This
states that for any volume tau surrounded by a closed
surface ,
In words 'The surface integral of any vector over a closed
surface S is equal to the divergence of that vector
integrated over the volume enclosed by S.' 
Stokes' theorem
This states
In words 'The line integral of any vector around a closed
path is equal to the surface integral of the curl of that
vector integrated over a surface S which is bounded by
the path of the line integral.'
THE END
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