Phys-272 Lecture 17
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Motional Electromotive Force (emf)
Induced Electric Fields
Displacement Currents
Maxwell’s Equations
From Faraday's Law to
Displacement Current
AC
generator
Magnetic Levitation Train
Review of Sources of B fields
Clicker
A long wire carries a current I.
The B field at distance r from the
wire
I
Question I
a)
B = µo I / 2π r 2
b)
B = µo Ir / 2π
c)
B = µo I / 2π r
Verify using
Ampere’s Law
Question II
Direction is
(a) radial, perp to wire (b) tangential
(c) along the wire
(d) anti-parallel to the wire
Motional Electromotive Force
d r r
dΦ B
ε = − ∫ B ⋅ dA = −
dt
dt
In Faraday’s Law, we can induce EMF in the loop when the
magnetic flux, ΦB, changes as a function of time. There are two
Cases when ΦB is changing,
1) Change the magnetic field (non-constant over time)
2) Change or move the loop in a constant magnetic field
The slide wire generator is an example of #2 and the induction of
EMF via moving parts of the loop is called, motional EMF.
Slide Wire Generator;
revisited again
Suppose we move a conducting bar in a constant B field, then
a force F=q v×B moves + charge up and – charge down. The charge
distribution produces an electric field and EMF, , between a & b.
This continues until equilibrium is reached.
r
r
r
r F qv × B r r
E= =
=v×B
q
q
br
r
ε = ∫ E ⋅ dl = vBl
a
In effect the bar’s motional EMF is an equivalent to a battery EMF
Slide Wire Generator;
revisited again
If the rod is on the U shaped conductor, the charges don’t build
up at the ends but move through the U shaped portion. They
produce an electric field in the circuit. The wire acts as a
source of EMF – just like a battery. Called motional
electromotive force.
r r
ε = ∫ E ⋅ dl = vBl
b
a
Direct Current Homopolar Generator invented by Faraday
Rotate a metal disk in a constant
perpendicular magnetic field. The
charges in the disk when moving
receive a radial force. The causes
current to flow from center to point
b.
R
1
2
ε = ∫ ωBr dr = ωBR 2
0
Faraday’s Law (continued)
What causes current to flow in wire?
Answer: an E field in the wire.
A changing magnetic flux not only causes
an EMF around a loop but an induced
electric field.
Can write Faraday’s Law:
r r
d r r
dΦ B
ε = ∫ E ⋅ dl = − ∫ B ⋅ dA = −
dt
dt
Remember for a long straight wire of length l, V = El.
Note: For electric fields from static charges, the EMF from a closed
path is always zero. Not true here.
There are two possible sources for electric fields !
Induced Electric Fields
Suppose we have electromagnetic
that has an increasing magnetic field
Using Faraday’s Law we predict,
N
r r
d r r
dΦ B
∫ E ⋅ dl = − ∫ B ⋅ dA = −
dt
dt
If we take a circular path inside and
centered on the magnet center axis,
the electric field will be tangent to the
circle. (E field lines are circles.) NOTE
such an E field can never be made by
static charges
B
E
S
E field lines will look like an onion slice
N.B. there are no wire loops, E fields can appear w/o loops
If we place a loop there, a current would flow in the loop
Induced Electric Fields; example
If we have a solenoid coil with changing current there will be circular
electric fields created outside the solenoid. It looks very much like the
mag. field around a current carrying wire, but it is an E field and there
are no wires or loops.
E
Note the E fields are
predicted by Faraday eqn.
r r
d r r
dΦ B
∫ E ⋅ dl = − ∫ B ⋅ dA = −
dt
dt
Eddy Currents
Changing magnetic fields in metal induce eddy currents.
Example: Energy loss in transformers.
To reduce use laminations.
But eddy currents often useful.
Maxwell’s Equations (integral form)
Name
Equation
Gauss’ Law for
Electricity
r r Q
∫ E ⋅ dA =
Gauss’ Law for
Magnetism
r r
∫ B ⋅ dA = 0
ε0
Faraday’s Law
r r
dΦ B
∫ E ⋅ dl = −
dt
Ampere’s Law
r
∫ B ⋅ dl = µ 0 i
Needs to be modified.
Description
Charge and
electric fields
Magnetic fields
Electrical effects
from changing B
field
Magnetic effects
from current
+?
There is a serious asymmetry.
Remarks on Gauss Law’s with different closed surfaces
r r Qenclosed
∫ E ⋅ dA =
r r
∫ B ⋅ dA = 0
ε0
Gauss Law’s works for
ANY CLOSED SURFACE
square
cylinder
Surfaces for
integration
of E flux
sphere
bagel
Remarks on Faraday’s Law with different attached surfaces
r r
r r
d ∫ B ⋅ dA
∫ E ⋅ dl = −
dt
Line integral
defines the
Closed loop
disk
Faraday’s Law works for
any closed Loop and ANY
attached surface area
Surface area
integration for B flux
cylinder
Fish bowl
This is proved in Vector Calculus with Stokes’ Theorem
Generalized Ampere’s Law and displacement current
r
∫ B ⋅ dl = µ0 I enclose , is incomplete.
Ampere’s original law,
Consider the parallel plate capacitor and suppose a current ic is
flowing charging up the plate. If Ampere’s law is applied for the
given path in either the plane surface or the bulging surface we
we should get the same results, but the bulging surface has ic=0,
so something is missing.
Generalized Ampere’s Law and displacement current
Maxwell solved dilemma by adding an addition term called
displacement current, iD = ε dΦE/dt, in analogy to Faraday’s Law.
r
dΦ E 

∫ B ⋅ dl = µ 0 (ic + iD ) = µ 0  ic + ε 0 dt 
Current is once more continuous: iD between the plates = iC in the wire.
q = CV
=
εA
( Ed )
d
= εEA = εΦ E
dq
dΦ E
= ic = ε
dt
dt
Summary of Faraday’s Law
r r
dΦ B
∫ E ⋅ dl = −
dt
If we form any closed loop, the
line integral of the electric field
equals the time rate change of
magnetic flux through the surface
enclosed by the loop.
B
If there is a changing magnetic field, then there will be
electric fields induced in closed paths. The electric fields
direction will tend to reduce the changing B field.
Note; it does not matter if there is a wire loop or an imaginary
closed path, an E field will be induced. Potential has
no meaning in this non-conservative E field.
E
Charge is flowing onto this parallel
plate capacitor at a rate dQ/dt=2 A
II
I
III
What is the displacement current in
regions I and III ?
A) 2 A
B) 1 A
C) 0
D) -2A
Charge is flowing onto this parallel
plate capacitor at a rate dQ/dt=2 A
II
I
III
What is the displacement current in
region II ?
A) -2/3A
B) 1 A
C) 2 A
D) O A
Summary of Ampere’s Generalized Law
r
dΦ E 


∫ B ⋅ dl = µ 0  ic + ε 0
dt 

Current ic
If we form any closed loop, the line
integral of the B field is nonzero if
there is (constant or changing) current
through the loop.
If there is a changing electric field
through the loop, then there will
be magnetic fields induced about a
closed loop path.
B
E
B
Maxwell’s Equations
James Clerk Maxwell (1831-1879)
• generalized Ampere’s Law
• made equations symmetric:
– a changing magnetic field produces an electric field
– a changing electric field produces a magnetic field
• Showed that Maxwell’s equations predicted
electromagnetic waves and c =1/√ε0µ0
• Unified electricity and magnetism and light.
All of electricity and magnetism can be summarized by
Maxwell’s Equations.
More important applications
of Faraday’s Law
Mutual Inductance
If we have a constant current i1 in coil 1,
a constant magnetic field is created and
this produces a constant magnetic flux in
coil 2. Since the ΦB2 is constant, there NO
induced current in coil 2.
If current i1 is time varying, then the ΦB2
flux is varying and this induces an emf ε2
in coil 2, the emf is
dΦ B2
ε 2 = −N2
dt
We introduce a ratio, called mutual inductance, of flux in coil 2
divided by the current in coil 1.
N 2Φ B 2
M 21 =
i1
Mutual Inductance
N 2Φ B 2
mutual inductance, M 21 =
, can now be used in Faraday’s eqn.
i1
M 21 i1 = N 2 Φ B 2
di1
dΦ B 2
di1
M 21
= N2
= −ε 2 ; ε 2 = − M 21
dt
dt
dt
We can also the varying current i2 which creates a changing flux ΦB1
in coil 1 and induces an emf ε1. This is given by a similar eqn.
di2
ε1 = − M12
dt
It can be shown (we do not prove here) that,
M 12 = M 21 = M
The units of mutual inductance is T ⋅ m2/A = Weber/A = Henry
(after Joseph Henry, who nearly discovered Faraday’s Law)
Mutual Inductance
The induced emf,
has the following features;
di2
ε 1 = −M
dt
• The induced emf opposes the magnetic flux change
(Lenz’s Law)
• The induced emf increases if the current changes very
fast
• The induced emf depends on M, which depends only the
geometry of the two coils and not the current.
• For a few simple cases, we can calculate M, but usually it
is just measured.
Problem 30.1
Two coils have mutual inductance of 3.25×10−4 H. The current
in the first coil increases at a uniform rate of 830 A/s .
A) What is the magnitude of induced emf in the 2nd coil? Is it
constant?
B) suppose that the current is instead in the 2nd coil, what is the
magnitude of the induced emf in the 1st coil?
di1
ε 2 = −M
dt
A
= −(3.25 × 10 H )(830 ) = − 0.27V
s
−4
di2
ε 1 = −M
= − 0.27 V
dt
Tesla Coil Example
Magnetic field due to coil 1 is
B1 = µ0n1i1 = µ0N1i1 / l
Mutual inductance is,
N 2 Φ B 2 N 2 B1 A
M=
=
i1
i1
N 2 µ0 N1i1 A µ0 N1N 2 A
=
=
i1l
l
The induced emf in coil1 from coil2 is
µ0 N1N 2 A di2
di2
ε1 = − M
=−
dt
l
dt