Electromagnetism
Physics 15b
Lecture #14
Induction
Purcell 7.1–7.5
What We Did Last Time
Moving charge near current receives force proportional to
its velocity


Identified as Lorentz force due to magnetic field
Electricity and magnetism are connected through relativity
Relativistic transformation of E and B fields
E′ = E
E′⊥ = γ (E⊥ + β × B ⊥ )
B′ = B B′⊥ = γ (B ⊥ − β × E⊥ )


Linear transformation
Consistent with the force transformation
F′ = F + F⊥ γ
1
Today’s Goals
Introduce electromagnetic induction
Changing magnetic field produces
electromotive force
  First step into non-static E&M

Discuss Faraday’s Law
First in integral form
  Magnetic flux  emf
  Then in differential form
  B field  E field
  Applications: AC power generator
  Eddy current

Michael Faraday (1791–1867)
Moving Rod in B
Move a conducting rod perpendicular to a uniform B field

Movable charge q in the rod receives Lorentz force
F=
q
v ×B
c
Net charges appear at the ends
  This in turn produces E field opposite
to v × B

B
++
++
F
E
q
––
––
v
Total force on charge q is
q
because current
Ftotal = v × B + qE = 0 Equilibrium
cannot
keep
flowing on this rod
c
1
E = − v × B must appear in the rod
c
2
Moving Loop in B
B
Now, move a rectangular conducting loop

Top and bottom bars feel the same effect
What happens when the loop goes
outside the B field?
No more Lorentz force in the bottom rod
  Top rod is still moving the charges to
produce E field
F q
+
+
+
+
+
+

Current I will flow around the loop
This is called electromagnetic induction
The current is induced by the loop’s motion
  Change of B is crucial

E
F q
E
F q
+
+
+
+
+
+
–
–
–
–
–
–
v
–
–
–
–
–
–
E
I
q
v
Faraday’s Law
Potential difference across the top bar is
vBw
Δφ = Ew =
c
x

It’s as if we have an emf in it
The emf appear only while the loop is
crossing the boundary of the B field

Consider the flux of B through the loop:
Φ≡

∫
loop
B ⋅ da = Bwx
B
F q
+
+
+
+
+
+
–
–
–
–
–
–
E
I
q
w
v
dΦ
dx
= Bw
= −Bwv
dt
dt
We can express the emf as
1 dΦ
E=−
Faraday’s Law
c dt
E=
vBw
c
3
Lenz’s Law
B
1 dΦ
What’s the minus sign in E = −
?
c dt

It comes out naturally if you define the flux
and emf consistently (w/ right-hand rule)
F q
x
Lenz’s Law helps you check the sign
+
+
+
+
+
+
E
I
q
–
–
–
–
–
–
The induced current creates additional
w
v
magnetic field that opposes changes of flux
  Since Φ is decreasing in this example,
the induced B has the same direction as the external B
  The induced current receives force from the magnetic field that
slows down the motion
  The magnetic force on the top bar points upward

This is an example of “Nature opposes changes”
Moving-Rod Circuit
A circuit is made of a movable metal rod on two rails
Rod moves with v
  Area of the circuit

A = xL
F
dA
= vL
dt
R
Flux is

Faraday’s law
This result in current I
E vBL
I= =
R cR
q
v
L
qE
dΦ
dA
= −B
= −BvL
dt
dt

B
x
E=−
1 dΦ BvL
=
c dt
c
The resistor dissipates
E 2 v 2B 2L2
IE = = 2
R
cR
Where did this energy com from?
4
Moving-Rod Circuit
Lorentz force on the induced current is
F=
I
L×B
c
I
Direction of F opposite to v
  To keep the bar moving,
we must pull the bar with
this much force to the right

B
R
v
F
L
x
ILB vB 2L2
F=
= 2
c
c R
How much work per unit time do we have to do?
v 2B 2L2
P = Fv = 2
= IE
c R
Exactly the power
dissipated in the resistor
Proof of Faraday’s Law
A loop of an arbitrary shape is moving with velocity v
through a static magnetic field B

Flux through the loop at time t and t + Δt
Φ(t) =
∫
S
B ⋅ da, Φ(t + Δt) =
∫
S′
Consider the volume enclosed by S, S′,
and the “ribbon” R between them

vΔt
S′
Since div B = 0, the total flux must be zero
0 = Φ(t) − Φ(t + Δt) + ∫ B ⋅ da
R

S
dL
B ⋅ da
∫
R
B ⋅ da = Φ(t + Δt) − Φ(t)
( )
Infinitesimal area da on the ribbon is da = vΔt × dL
∫
R
B ⋅ da = 
∫
loop
B ⋅ (vΔt × dL)
dΦ
= ∫ B ⋅ (v × dL)
loop
dt 
5
Proof of Faraday’s Law
dΦ
= ∫ B ⋅ (v × dL)
loop
dt 

Use math identity:
a ⋅ (b × c) = b ⋅ (c × a) = c ⋅ (a × b)
dΦ
= ∫ dL ⋅ (B × v) = − ∫ (v × B) ⋅ dL
dt loop
loop
dL
vΔt
S
S′
Imagine a unit charge on, and moving with, the loop

The Lorentz force acting on it is v×B
The loop integral represents the work the Lorentz force
would do if a unit charge were moved around the wire,
i.e. the emf
1 dΦ
E=−
c dt
Relativity
Faraday’s law says “flux changes  emf happens”

It doesn’t say why the flux changes
What if B field itself changes while the loop is static?

Relativity: we should get the same result
  Same problem in another reference frame
Test this experimentally
A

S
N
A
Current is generated in a loop of wire when
  Magnet approaches
  Current flows in a nearby wire
6
Ways to Change Flux
Magnetic flux Φ depends on the B field, the size, shape
and angle of the loop
A
  Simple case: a flat loop of
θ
area A in a uniform B field
B
Φ=
∫ B ⋅ d A = BA cos θ
E=−
1 dΦ
1d
=−
(BA cos θ )
c dt
c dt
Induction may occur because of
Changing B field
  Changing area A of the loop
  Changing angle θ between B and the loop

AC Power Generator
Alternate Current (AC) generators are very simple

For a loop area A rotating
with angular velocity ω
Φ = BA cos θ = BA cos ω t
1 dΦB BAω
E=−
=
sin ω t
c dt
c

If the loop has N turns
E=

NBAω
sin ω t
c
For commercial 60 Hz
power generator
ω = 2π f = 120π sec
7
Eddy Currents
Faraday’s Law works in conductor of any shape
Consider a simple plate
Increase B field  Rotating current
  Move the plate into a B field  Ditto


Rotating current in a continuous body
of conductor due to changing B field
is called the eddy current

Direction is given by Lenz’s Law
Eddy currents always slow down the change

Used for braking systems of various machines
Differential Form of Faraday
Faraday’s law in integral form: E = −

The emf can be expressed as
E=
∫ E ⋅ds =
−
∫
S
1 dΦ
c dt
(∇ × E) ⋅ da
1 dΦ
1d
=−
c dt
c dt
∫
B ⋅ da

RHS is

Apply this to an arbitrary, but stationary, surface S
lhs − rhs =
S
⎛
1 ∂B ⎞
∇×E+
⋅ da = 0
⎜
S⎝
c ∂t ⎟⎠
∫
∇×E = −
1 ∂B
c ∂t
curl E is no longer zero — Leaving electrostatics
8
Maxwell’s Equations
All the equations in differential form that we found so far:
⎧∇ ⋅E = 4πρ
⎪
⎪∇ ⋅B = 0
⎪
1 ∂B
⎨∇ × E = −
c ∂t
⎪
⎪
4π
J
⎪∇ × B =
c
⎩
 Relates E and charge density ρ — Gauss
 No magnetic monopoles
 Change in B creates E — Faraday
 Relates B and current density J — Ampere
Another step toward Maxwell’s equations
One last term is missing — Where is it?


Hint #1: Symmetry
Hint #2: Look at the Lorentz transformation of fields
Summary
Induction: emf when the magnetic flux in a loop changes
B
1 dΦ
Faraday’s Law E = −
FL q
c dt
–
+

Sign of the emf follows Lenz’s Law:
the induced current opposes the change
of the flux
Differential form: ∇ × E = −
1 ∂B
c ∂t

curl E no longer zero!

Just one last step before completing
Maxwell’s equations
x
+
+
+
+
+
–
–
–
–
–
E
I
q
w
E=
v
vwB
c
9