# JS 3010 Electromagnetism I

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16 Νοε 2013 (πριν από 4 χρόνια και 5 μήνες)

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JS 3010 Electromagnetism I

Dr. Charles Patterson

2.48 Lloyd Building

Course Outline

Course texts:

Electromagnetism
, 2nd Edn. Grant and Phillips (Wiley)

Electromagnetic Fields and Waves
, 2nd Edn. Lorrain and
Corson (Freeman)

Online at:
www.tcd.ie/Physics/People/Charles.Patterson/Teaching/JS/JS3010/

Topics:

0). Overview

1). Vector Operators and Vector Analysis

2). Gauss’ Law and applications

3). Electrostatic and Dielectric Phenomena

4). Ampere’s Law and Applications

5). Magnetostatic and Magnetic Phenomena

6). Maxwell’s Equations and Electromagnetic Radiation

Lorentz force on single charge q

F
B

= q
v

x
B

B

magnetic induction (Tesla, T)

F
E

= q
E

E
electric field strength (Volts/m)

Force on charge due to electric
and magnetic fields

i

j

k

v

B

F

i x j = k

i

j

k

E

F

Sense of
F

depends on sign of q

Electric field strength
E
(
r
,t) Volts m
-
1

or NC
-
1

Vector field
of position and time

Field at
field point

r

due to single point charge at
source
point

r’

(electric monopole)

Note
r
-
r’

vector directed away from source point when q
is positive. Electric field lines point away from (towards) a
positive (negative) charge

Electric Fields

r'
-
r
r'
-
r
r
E
3
o
q
4
1
)
(


O

r

r’

r
-
r’

Magnetic Fields

Magnetic Induction (Magnetic flux density)
B
(
r
,t) Tesla (T)
Vector field of position and time

Field at
field point

r

due to current element at
source point

r’
is given by Biot
-
Savart Law

Note d
B
(
r
) is the contribution to the circulating magnetic
field which surrounds this infinite wire from the current
element d
l

3
o
)
x(
dl'
4
)
(
d
r'
r
r'
r
r
B

O

r

r’

r
-
r’

d
B
(
r
)

dl’

Maxwell’s Equations

Expressed in integral or differential forms

Simplest to derive integral form from physical principle

Equations easier to use in differential form

Forms related by vector field identities (Stokes’ Theorem,
Gauss’ Divergence Theorem)

Time
-
independent problems electrostatics, magnetostatics

Time
-
dependent problems electromagnetic waves

t
x
t
x
.
.
f
f

D
j
H
B
E
B
D
0

Vacuum Matter

t
c
1
x
t
x
.
.
2
o
o

E
j
B
B
E
B
E

0
1). Vector Operators and Analysis

Div, Grad, Curl (and all that)

Del or nabla operator

In Cartesian coordinates

Combining vectors in 3 ways

Scalar (inner) product

a
.
b

=
c

(scalar)

Cross (vector) product

a
x
b

=
c

(vector)

Outer product (dyad)

ab

=
c

(tensor)

z
y
x
,
,
Scalar Product
-

Divergence

r
is a Cartesian position vector

r
=
(x,y,z)

A

is vector function of position
r

Div

A
=

Scalar product of
del

with
A

Scalar function of position

z
A
y
A
x
A
.
z
y
x

A

z
y
x
A
,
A
,
A
)
(

r
A
Cross Product
-

Curl

Curl
A

=

Cross product of
del

with
A

Vector function of position

y
A
x
A
z
A
x
A
z
A
y
A
x
x
y
x
z
y
z
k
j
i
A
z
y
x
A
A
A
z
y
x
k
j
i

A
x
i

j

k

Gradient

f
(x,y,z)

is a scalar function of position

Grad
f

=

f 

Operation of del on scalar function

Vector function of position

z
φ
,
y
φ
,
x
φ
f
=const.

f

Div Grad

the Laplacian

Inner product Del squared

Operates on a scalar function to produce a scalar
function

Outer product

2
2
2
2
2
2
z
y
x
.


2
2
2
2
2
2
2
2
2
2
2
2
z
y
z
x
z
z
y
y
x
y
z
x
y
x
x
Green’s Theorem on plane

Leads to Divergence Theorem and Stokes’ Theorem

Fundamental theorem of calculus

Green’s Theorem

f(a)
f(b)
f(x)dx
dx
d
b
a

dxdy
y
y)
P(x,
x
y)
Q(x,
y)dy
Q(x,
y)dx
P(x,
A
C


P(x,y), Q(x,y) functions with

continuous partial derivatives

a

b

c

d

area A

contour C

x

y

Green’s Theorem on plane

Integral of derivative over A

Integral around contour C



d
c
b
a
d
c
A
dy
y)
Q(a,
-
y)
Q(b,
dx
x
y)
Q(x,
dy
dxdy
x
y)
Q(x,

a

b

c

d

Area A

contour C

x

y

dy
y)
Q(a,
y)
Q(b,
y)dy
Q(a,
y)dy
Q(b,
y)dy
Q(x,
d
c
c
d
d
c
C

dxdy
x
y)
Q(x,
y)dy
Q(x,
A
C


Green’s Theorem on plane

Similarly

Green’s Theorem relates an integral along a
closed

contour C
to an area integral over the
enclosed

area A

QED for a rectangular area (previous slide)

Consider
two

rectangles and then
arbitrary planar surface

Green’s Theorem applies to arbitrary, bounded surfaces

dxdy
y
y)
P(x,
y)dx
P(x,
A
C


=

Contributions from boundaries cancel

cancellation

No cancellation on boundary

C

A

Divergence Theorem

Tangent d
r

=
i

dx +
j

dy

Outward normal
n
ds =
i

dy

j

dx

n

unit vector along outward normal

ds = (dx
2
+dy
2
)
1/2

P(x,y) =
-
V
y

Q(x,y) = V
x

Cartesian components of the same vector field
V

Pdx + Qdy =
-
V
y
dx + V
x
dy

(
i

V
x

+
j

V
y
).(
i

dy

j

dx) =
-
V
y

dx + V
x

dy =
V
.
n

ds

i

j

d
r

n
ds

dx

dy

dx

dy

V

= (V
x
,V
y
)

Divergence Theorem 2
-
D 3
-
D

Apply Green’s Theorem

In words

-

Integral of
V
.
n

ds over surface contour equals
integral of div
V

over surface area

In 3
-
D

Integral of
V
.
n

dS over bounding surface S equals integral
of div
V

dv within volume enclosed by surface S

dxdy

.
dxdy
y
V
x
V
ds

.
dxdy
y
y)
P(x,
x
y)
Q(x,
y)dy
Q(x,
y)dx
P(x,
A
A
y
x
C
A
C





V
n
V

V
S
dv

.
dS
.
V

n
V
V
.
n

dS

.
V

dv

Curl and Stokes’ Theorem

For divergence theorem P(x,y) =
-
V
y

Q(x,y) = V
x

Instead choose

P(x,y) = V
x

Q(x,y) = V
y

Pdx + Qdy = V
x
dx + V
y
dy

V

=
i

V
x

+
j

V
y

+ 0
k

dxdy

.

x
.d

.

x
y
V
x
V
y
y)
P(x,
x
y)
Q(x,
d

.
dy

dx
V

V
y)dy
Q(x,
y)dx
P(x,
dy

V
dx

V
y)
Q(x,
y)dx
P(x,
A
C
x
y
y
x
y
x


k
V
r
V
k
V
r
V
j

i
j

i

.

d
r

dx

dy

V

= (V
x
,V
y
)

i

k

j

local value of

x
V

A

C

Stokes’ Theorem 3
-
D

In words

-

Integral of (

x
V)
.
n

dS over surface S equals
integral of
V.
d
r

over bounding contour C

It doesn’t matter which surface (blue or hatched). Direction
of d
r

determined by right hand rule.

(

x
V)
.
n

dS

n

outward normal

dS

local value of

x
V

local value of
V

d
r

V
.

d
r

dS

.

x
.d
S
C


n
V
r
V

S

C

Summary

Green’s Theorem

Divergence theorem

Stokes’ Theorem

Continuity equation

dxdy
y
y)
P(x,
x
y)
Q(x,
y)dy
Q(x,
y)dx
P(x,
A
C


V
S
dv

.
dS
.
A

n
V

dS

.

x
.d
S
C


n
V
r
V

0
t
t)
,
ρ(
t)
,
(
.

r
r
j
V
.
n

dS

.
A

dv

(

x
V)
.
n

dS

n

outward normal

dS

local value of

x
V

local value of
V

d
r

V
.

d
r

S

C

surface S

volume v