1
Coulombs Law and Electric Field Intensity
C OULOMB S L AW
Force (a vector) on one charge due to another charge is:
proportional to the product of the charges
inversely proportional to the square of the distance between them
in a direction parallel to th

Gauss Law for Symmetrical Charge
Distributions - Step by Step
8 November 2011
=
I
DdS
Although Gauss law is always true, it is only particularly useful in cases of symmetric charge
distributions. It is worth discussing considerations for symmetry rst.

Calculating Scalar Potential
15 October 2012
ZZ Z
)=
(r
0
4 r
r
1. Dene an origin at a convenient point
2. Write the vector 0 from the origin to the source (charge) location, in an appropriate
r
coordinate system (Note: if you have a continuous distri

Wednesday Tutorial #1: Vector Fundamentals
1
Position Vectors
The vector V denes the position of point P in the coordinate system O, where
V = 5 ax 3 ay + 12 az .
Consider a dierent coordinate system O , with its origin located at
O = 9 ax + 1 ay 5 az ,
i

1. A vector eld throughout free space is given in cylindrical coordinates as
V
F = 2 2 + 1 cos b 2 + 1 sin b + 22 cos b
a
a
a
m
Consider the contour which follows the path (and in the direction)
as shown in the gure below
(a) Evaluate the integral
Z

Wednesday Tutorial # 2:
The Electric Field from Two Point Charges
1
Electric Field From Two Opposite Charges
(a.k.a. The Electric Dipole)
A point charge of +q is placed at z = +d/2 and a point charge of q is placed at z = d/2.
Find:
a) The electric eld ev

Wednesday Tutorial #3:
Gauss Law and a Coaxial Line
Consider an innite coaxial line with inner radius a and outer radius b. The inner conductor has a surface charge density of +s and the outer conductor has a surface charge of
-a/b s .
Find the electric e

Vector Algebra Supplemental Examples
Vector Algebra
Example 1 If in rectangular coordinates three vectors are f = (1, 3, 2) , b = (1, 1, 1) , c = (2, 6, 9)
nd
i) |f + c|
Ans:
f + c = (1, 3, 2) + (2, 6, 9)
b
f
ii) |f | + 5 |b|
Ans: Since in general
i

Maxwells Equations
integral form
ZZ
~ ~
~
~
E dL =
B dS
ZZ ~
I
D
~
~
~
dS
H dL = +
ZZZ
I
~
~
D dS = =
I
I
point form
~
B
~
~
E =
~
~
~ D
H =J +
Faradays Law
Amperes Law
Gauss Law for Electric Fields
~ ~
D =
Gauss Law for Magnetic Fields
~ ~
B =0
~
~

1
Electric Potential and Energy
E NERGY E XPENDED M OVING
P OINT C HARGE IN E LECTRIC
F IELD
=
Z
P OTENTIAL D IFFERENCE
=
(~) =
r
ZZZ
0
~ ~
EdL
Z
P OTENTIAL F IELD OF A C HARGE
D ISTRIBUTION
~ ~
EdL
Found by evaluating the work done in moving a uni

Assume a cloud of electrons is conned in a region of space between
two spheres of radius 1 = 2 cm and 2 = 5 cm with a charge density of
8
C
= 310 cos2 m2 . Find the total charge in the cloud
4
Ans:
ZZZ
( = 2 sin )
=
=
2
Z
0
Z
0
005
Z
2
3 108
2
sin

Example 1 A sphere of radius r = a contains a uniform charge distribution of v
vector everywhere in space by using Gauss Law.
C
m2 .
Determine the electric ux density
Ans: First consider arguments of symmetry. Centre the sphere at the origin of a spherica

ENEL475 - Electromagnetic Fields - Fall 2014
Electric Potential - Practice Problem Solutions
Page: 1
ENEL475 - Electromagnetic Fields - Fall 2014
Page: 2

Example 1 (Field from a Circle of Charge) Consider a circle of radius , lying in the C
plane, centred at the origin. The circle has a uniform linear charge distribution m as shown in
~
the gure below. Find the electric eld E at any point on the -axis = (0

1
Electric Materials and Boundary Conditions
C URRENT AND C URRENT
D ENSITY
=
=
ZZ
~ ~
J dS
Electric charges in motion constitute current which is a
scalar quantity (the amplitude of the current). The total
current crossing a surface then depends on the c

Total Flux Calculations - Examples
C
~
Example 1 Given D=10b + 5b 2 , nd the electric ux passing through the surface dened by
a
a
m
= 6 m,0 90 2 2 m
Ans: First we must identify the pictorial representation of the surface. In this
case, the radius is xed,

Coulombs Law
7 November 2011
1. Dene an origin at a convenient point
2. Write the vector 0 from the origin to the source (charge) location (Note: if you have a
r
continuous distribution of sources, i.e. a charge density, then this vector is to some genera