JS 3010 Electromagnetism I

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16 Νοε 2013 (πριν από 4 χρόνια και 1 μήνα)

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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