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/
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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
Force on charge due to electric
and magnetic fields
• Lorentz force on single charge q
FB = q v x B
B magnetic induction (Tesla, T)
ixj=k
i
k
v
j
j
B
F
FE = q E
i
k
E electric field strength (Volts/m)
Sense of F depends on sign of q
E
F
Electric Fields
• 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)
E(r ) 
1
q
4o r - r'
3
r - r' 
r-r’
r
r’
O
Note r-r’ vector directed away from source point when q
is positive. Electric field lines point away from (towards) a
positive (negative) charge
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

dB(r)
o  dl' x(r  r' )
dB(r ) 
4 r  r' 3
r
O
r’
r-r’
dl’
• Note dB(r) is the contribution to the circulating magnetic
field which surrounds this infinite wire from the current
element dl’
.E 
Maxwell’s
Equations

o
.B  0
.B  0
B
xE  
t
xE  
1 E
xB  o j  2
c t
Vacuum
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•
.D   f
B
t
D
xH  jf 
t
Matter
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
1). Vector Operators and Analysis
• Div, Grad, Curl (and all that)
    
• Del or nabla operator
   , , 
 x y z 
– In Cartesian coordinates
• Combining vectors in 3 ways
– Scalar (inner) product
– Cross (vector) product
– Outer product (dyad)
a.b
axb
ab
= c (scalar)
= c (vector)
= c (tensor)
Scalar Product - Divergence
• r is a Cartesian position vector r=(x,y,z)
• A is vector function of position r A(r )  A x , A y , A z 
A x A y A z


• Div A = .A 
x
y
z
• Scalar product of del with A
• Scalar function of position
Cross Product - Curl
i


x
A

• Curl A =
x
Ax
j

y
Ay
k

z
Az
 A z A y 
 A z A x 
  j 
xA  i 


 k
z 
z 
 x
 y
• Cross product of del with A
• Vector function of position
k
j
i
 A y A x 



y 
 x
Gradient
 f(x,y,z) is a scalar function of position
 φ φ φ 
• Grad f = f   , , 
 x y z 
• Operation of del on scalar function
• Vector function of position

f
f =const.
Div Grad – the Laplacian
2
2
2
• Inner product Del squared .  2  2  2
x
y
z
• Operates on a scalar function to produce a scalar
function
 2
 2
 x
 2
• Outer product   
 y2 x
 
 zx

2
xy
2
y 2
2
zy
2 

xz 
2 

yz 
2 
z 2 
Green’s Theorem on plane
• Leads to Divergence Theorem and Stokes’ Theorem
• Fundamental theorem of calculus
y
b
d
a dxf(x)dx  f(b)  f(a)
d
P(x,y), Q(x,y) functions with
continuous partial derivatives
area A
contour C
c
x
a
b
 Q(x, y) P(x, y) 
dxdy

• Green’s Theorem  P(x, y)dx  Q(x, y)dy  A 
y 
 x
C
Green’s Theorem on plane
• Integral of derivative over A
d
b
Q(x, y)
Q(x, y)
dxdy

dy
A x
c a x dx
d
  Q(b, y) - Q(a, y) dy
c
• Integral around contour C
c
c
d
 Q(x, y)dy   Q(b, y)dy   Q(a, y)dy
C
y
d
d
  Q(b, y)  Q(a, y) dy
d
c
Area A
contour C
c
x
a
b
Q(x, y)
C Q(x, y)dy  A x dxdy
Green’s Theorem on plane
• Similarly
P(x, y)
C P(x, y)dx  A y dxdy
• 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
=
cancellation
A
C
Contributions from boundaries cancel
No cancellation on boundary
• Green’s Theorem applies to arbitrary, bounded surfaces
Divergence Theorem
j
V = (Vx,Vy)
nds
i
Tangent dr = i dx + j dy
dy
Outward normal n ds = i dy – j dx
n unit vector along outward normal
ds = (dx2+dy2)1/2
P(x,y) = -Vy
Q(x,y) = Vx
Cartesian components of the same vector field V
• Pdx + Qdy = -Vydx + Vxdy
• (i Vx + j Vy).(i dy – j dx) = -Vy dx + Vx dy = V.n ds
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•
dx
dr
dx
dy
Divergence Theorem 2-D 3-D
• Apply Green’s Theorem
 Q(x, y) P(x, y) 
P(x,
y)dx

Q(x,
y)dy

C
A  x  y dxdy
 Vx Vy 
C V.n ds  A  x  y dxdy  A.V dxdy
• In words - Integral of V.n ds over surface contour equals
integral of div V over surface area
V.n dS
.V dv
• In 3-D  V.n dS  V.V dv
S
• Integral of V.n dS over bounding surface S equals integral
of div V dv within volume enclosed by surface S
Curl and Stokes’ Theorem
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For divergence theorem P(x,y) = -Vy Q(x,y) = Vx
Instead choose
P(x,y) = Vx Q(x,y) = Vy
Pdx + Qdy = Vx dx + Vy dy
V = i Vx + j Vy + 0 k
P(x, y)dx  Q(x, y)  Vx dx  Vy dy
P(x, y)dx  Q(x, y)dy  i Vx  j Vy . i dx  j dy   V . dr
Q(x, y) P(x, y) Vy Vx



  x V  . k
x
y
x
y
 V.dr    x V  . k dxdy
C
k
local value of  x V
A
j
i
A
V = (Vx,Vy)
dx
C
dr
dy
Stokes’ Theorem 3-D
• In words - Integral of ( x V) .n dS over surface S equals
integral of V.dr over bounding contour C
• It doesn’t matter which surface (blue or hatched). Direction
of dr determined by right hand rule.
local value of  x V
n outward normal
S
dr
C
dS ( x V) .n dS
V. dr
 V.dr    x V  . n dS
C
local value of V
S
Summary
• Green’s Theorem
 Q(x, y) P(x, y) 
C P(x, y)dx  Q(x, y)dy  A  x  y dxdy
• Divergence theorem
surface S
V.n dS
 V.n dS   .A dv
V
S
• Stokes’ Theorem
S
 V.dr    x V  . n dS
C
volume v
local value of  x V
n outward normal
dS ( x V) .n dS
S
• Continuity equation
. j(r, t) 
.A dv
ρ(r, t)
0
t
dr
C
V. dr
local value of V