Energy stored in electromagnetic field
We#found#the#work#needed#to#assemble#a#static#charge#configuration#(against#the#
Coulombic#interaction)#is
I
✏0
2
E d⌧,
Welec =
2 (1)
where#E"is#the#resulting#electric#field.#The#above#is#also#interpreted#as#the#energy#stored#in#
the#electric#field.
Similarly,#we#found#that#the#work#required#to#flow#the#currents#(against#the#back#emf)#is
I
1
W =
2µ0
B 2 d⌧,
(1)
where#B"is#the#resulting#magnetic#field.#This#can#also#be#interpreted#as#the#energy#stored#in#
the#magnetic#field.
Therefore,#we#can#conclude#that#the#total#energy#stored#in#electromagnetic#fields#is#
Uem
1
=
2
I
(1)
✓
1 2
✏0 E + B
µ0
2
◆
d⌧.
Energy conservation law for electrodynamics
Consider#an#arbitrary#charge#and#current#configuration#which#produces#fields#E"and#B"at#time#
t.#Suppose#in#the#next#instant,#dt,#the#charges#move#around#a#bit.#We#would#like#to#know#how#
much#work,#dW,#is#done#the#the#electromagnetic#forces#acting#on#these#charges#in#the#interval#
dt.
Lorentz#force#law#tells#us#that#the#work#done#on#a#charge#q#is#
F · dl = q(E + v ⇥ B) · vdt = qE · vdt.
If#ρ#be#the#volume#charge#density,#then#work#done#on#all#charges#contained#in#volume#V#in#
Z
time#dt#is
dW =
⇢d⌧ E · vdt
V
Z
Z
dW
)
=
E · (⇢v) d⌧ =
E · J d⌧.
dt
V
V
From#above#equation,#we#infer#that#EIJ"#is#the#work#done#per#unit#time,#per#unit#volume,#i.e.,#it#
is#the#power#delivered#per#unit#volume.#Our#next#goal#is#to#express#this#quantity#in#terms#of#
fields#alone.#We#eliminate#J#with#the#aid#of#Ampere’s#circuital#law
1
E·J=E·
µ0
✓
r⇥B
@E
µ0 ✏ 0
@t
◆
1
= E · (r ⇥ B)
µ0
@E
✏0 E ·
@t
Energy conservation law for electrodynamics
1
E·J=E·
µ0
✓
r⇥B
@E
µ0 ✏ 0
@t
◆
1
= E · (r ⇥ B)
µ0
@E
✏0 E ·
@t
Now#we#use#the#the#product#rule#for#divergence#of#crossOproduct#of#two#vectors#to#obtain
r · (E ⇥ B) = B · (r ⇥ E) E · (r ⇥ B)
) E · (r ⇥ B) = B · (r ⇥ E) r · (E ⇥ B).
Using#Faraday’s#law#for#curl#of#E,#we#obtain
E · (r ⇥ B) =
@B
B·
@t
r · (E ⇥ B).
Consequently,
E·J=
@E
✏0 E ·
@t
1
@B
B·
µ0
@t
1
r · (E ⇥ B).
µ0
Energy conservation law for electrodynamics
E·J=
@E
✏0 E ·
@t
1
@B
B·
µ0
@t
1
r · (E ⇥ B).
µ0
Now,#note#that
✓
1@ 2 1@
1
E =
(E · E) =
2 @t
2 @t
2
Similarly,
1@ 2 1@
1
B =
(B · B) =
2 @t
2 @t
2
Therefore,
E·J=
1@
2 @t
✓
@E @E
E·
+
·E
@t
@t
✓
@B @B
B·
+
·B
@t
@t
1 2
✏0 E + B
µ0
2
◆
◆
@E
=E·
@t
◆
@B
=B·
@t
1
r · (E ⇥ B).
µ0
Energy conservation law for electrodynamics
E·J=
Therefore,
1@
2 @t
✓
1 2
✏0 E + B
µ0
2
◆
1
r · (E ⇥ B).
µ0
Z
dW
E · J d⌧
=
dt
V
✓
◆
Z 
dW
1@
1 2
2
)
=
✏0 E + B
d⌧
dt
2 @t
µ0
V
1
µ0
Z
V
r · (E ⇥ B)d⌧.
On#RHS,#we#now#flip#the#ordering#of##volumeOintegration#and#timeOderivative#operation#in#first#
term,#and#apply#divergence#theorem#in#the#second#term#to#convert#it#to#a#surface#integral:#
dW
=
dt
d
dt
Z
V
1
2
✓
1 2
✏0 E + B
µ0
2
◆
d⌧
1
µ0
I
S
(E ⇥ B) · da.
Here ! is the surface enclosing the volume ", and da represents the area element.
dW
=
dt
d
dt
Z
Poynting’s theorem
V
1
2
✓
1 2
✏0 E + B
µ0
2
◆
d⌧
1
µ0
I
S
(E ⇥ B) · da.
The#above#relation#depicts#Poynting’s#theorem#in#mathematical#form.#It#is#the#“workO
energy”#theorem#of#electrodynamics.#The#first#integral#on#the#RHS#is#the#total#energy#
contained#in#the#electromagnetic#field.#Therefore,#its#−ve#timeOrate#of#change#gives#the#
information#about#loss#of#this#energy#from#within#the#volume#with#time.#The#second#term#
represents#the#rate#at#which#energy#is#flows#out#of# ,#across#its#boundary#surface# ,#by#the#
electromagnetic#fields.
Poynting#theorem#says#that,#the#work#done#on#the#charges#by#the#electromagnetic#force#is#
equal#to#the#decrease#in#energy#stored#in#the#field,#less#the#energy#that#flowed#out#through#
the#surface.
The#energy#per#unit#time,#per#unit#area,#transported#by#the#fields#is#called#the#Poynting"
vector:
1
S = (E ⇥ B).
µ0
(Be careful of the notational change. We are now using da for area element, and S for Poynting vector.)
Image Source: Wikipedia
https://en.wikipedia.org
/wiki/Poynting_vector
A simple DC circuit consisting of a battery (V) and a resistor (R), showing the Poynting
vectors (S, blue arrows) in the space surrounding it, as well as the fields it is derived
from, the electric field (E, red arrows) and the magnetic field (H,
green arrows). P = E x H The Poynting vector represents the direction and magnitude of
the power flow in the electromagnetic field (the length of the vectors shown here are not
to scale; only the direction is being shown) In the region of space around the battery, the
Poynting vectors are directed outward, indicating that power flows out from the battery
into the electromagnetic field. In the region of space around the resistor, the Poynting
vectors are directed inward, indicating that since the resistor consumes power, the
power enters it from the field. On any plane (P) located between the battery and the
resistor, it can be seen that the power flux though the plane is directed toward the
resistor.
Poynting vector in a coaxial
cable, shown in red crosses
Image Source: Wikipedia
https://en.wikipedia.org/wiki/Poyntin
g_vector
Poynting vector within the dielectric insulator of a coaxial cable is nearly parallel
to the wire axis (assuming no fields outside the cable and a wavelength longer
than the diameter of the cable, including DC). Electrical energy delivered to the
load is flowing entirely through the dielectric between the conductors. Very little
energy flows in the conductors themselves, since the electric field strength is
nearly zero. The energy flowing in the conductors flows radially into the
conductors and accounts for energy lost to resistive heating of the conductor.
No energy flows outside the cable, either, since there the magnetic fields of
inner and outer conductors cancel to zero.
WAVES: Basics
A# wave#is#usually# associated#with# a#disturbance# of# a#continuous#medium# that# propagates#
with#a#fixed#shape#at#constant#velocity.#
However,#
if#absorption#(loss)#is#present,#the#wave#diminishes#in#size#as#it#moves.
if#the#medium#is#dispersive,#different#frequencies#travel#at#different#speeds.
in#two#or#three#dimensions,#as#the#wave#spreads#out#its#amplitude#decreases.
standing#waves#don’t#propagate#at#all.
some# waves# don’t# require# any# medium# to# propagate.# In# such# cases# some# oscillatory#
fields#can#travel#through#vacuum#(We’ll#be#encounter#these#in#a#few#slides)
WAVES: Basics
Let#us#start#with#the#simplest#case,#fixed#shape#and#constant#speed#(v).#
Let# us#examine# the#wave#shown# at# two#different# times,# one#at# t=0,# and# again# at# some# later#
time#t.#Each#point#on#the#wave#form#simply#shifts#to#the#right#by#an#amount#vt.
For#example,#the#wave#can#be#generated#by#shaking#one#end#of#a#taut#string,#and#let#f(z,t)#be#
the#displacement#of# the#string#at# point# z,#and#at#time#t.# Given#the#initial#shape#of# the#string#
f(z,0)#≡#g(z).#Not#in#time#t#the#same#shape#would#have#traveled#a#distance#of#vt.#This#means#the#
wave#at#position#z#at#time#t#would#look#like#what#it#looked#like#at#z−vt#at#time#0,#i.e.,
f (z, t) = f (z
vt, 0) ⌘ g(z
vt)
#Therefore,#we#conclude#that#for#waves,#the#structure#involving#z#and#vt#must#appear#as#z−vt.#
Thus,#mathematically#the#waves#will#involve#have#functional#form#f(z, t)=g(z−vt).
The Classical wave equation
The#wave# equation,# which#gives# the#evolution# of# disturbance#f# as#a#function#of# z# and#t# is#
given#by
2
2
@ f (z, t)
1 @ f (z, t)
=
.
@z 2
v 2 @t2
Here#v#is#the#speed#of#propagation#along#the#zOdirection.
The#above#equation#is#referred#to# as#the#wave#equation,#because#it# admits#as#solutions#all#
functions# of# the# form# f(z,t)# ≡# g(z−vt),# i.e.,# the# waves# propagating# in# the# zOdirection# with#
speed#v.
Note# that# the# wave# equation# also# admits# solutions# of# the# form# f(z,t)# ≡# g(z+vt),# which#
represent#waves#moving#in#the#negative#z#direction#with#speed#v.
The#general#solution#is
f (z, t) = g(z
vt) + h(z + vt).
This# last# statement# is# consequence# of# the# fact# wave# equation# is# linear.# The# sum# (or# in#
general#a#linear#combination)#of#two#solutions#is#also#a#solution.#Every#solution#to#the#wave#
equation#can#be#expressed#in#this#form.