# Mutual inductance ```Mutual inductance
z
z
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Mutual induction –
current in one coil
induces emf in other coil
Distinguish from
self-induction
Mutual inductance, M21
of coil 2 with respect to
coil 1 is
NΦ B
L=
i
M 21
N 2 Φ 21
=
i1
Mutual inductance
M 21
z
N 2 Φ 21
=
i1
Rearrange equation
M 21i1 = N 2Φ 21
z
z
ε
z
2
z
dΦ 21
= −N2
dt
Induced emf in coil 2
due to i in coil 1 is
ε
Vary i1 with time
dΦ 21
di1
M 21
= N2
dt
dt
2
di1
= − M 21
dt
Obeys Lenz’s law
(minus sign)
Mutual inductance
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Reverse roles of coils
What is induced emf in
coil 1 from a changing
current in coil 2?
Same game as before
M 12
ε
1
N 1Φ 12
=
i2
di2
= − M 12
dt
Mutual inductance
z
The mutual inductance terms are equal
M 12
z
N 1Φ 12
=
i2
M 21 = M 12 = M
Rewrite emfs as
ε
z
M 21
N 2 Φ 21
=
i1
2
di1
= −M
dt
ε
1
di2
= −M
dt
Notice same form as self-induced emf
ε
L
di
= −L
dt
NΦ B
L=
i
Induced Electric Fields
z
Put a copper ring in a
uniform B field which is
increasing in time so the
magnetic flux through the
copper ring is changing
z
emf and current are
produced
z
If there is a current there
must be an E field present
to move the conduction
electrons around ring
Induced Electric Fields
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z
z
Induced E field acts the same way
as an E field produced by static
charges, it will exert a force, F=qE,
on a charged particle
True even if there is no copper ring
(the picture shows a region of
magnetic field increasing into the
board which produces circular
electric field lines).
Restate Faraday’s law – A changing
B field produces an E field given by
r r
dΦ B
∫ E • ds = − dt
No magnetic monopoles
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z
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Magnetic monopoles do not exist
Express mathematically as
r r
Φ B = ∫ B • dA = 0
Integral is taken over
closed surface
Net magnetic flux through
closed surface is zero
z
As many B field lines enter
as leave the surface
Electric charge x no magnetic charge
z
Gauss’s law for E fields
r r qenc
Φ E = ∫ E • dA =
ε0
z
Gauss’s law for B fields
r r
Φ B = ∫ B • dA = 0
z
Both cases integrate over closed Gaussian
surface
z
field is induced along a
r r
dΦ B
closed loop by a changing
E • ds = −
dt
magnetic flux encircled by
that loop
Is the reverse true?
Maxwell’s law of induction
B field is induced along a
r r
dΦ E
closed loop by a changing
B • ds = μ0ε 0
dt
electric flux in region
encircled by loop
∫
z
z
∫
Ampere’s + Maxwell’s law
r r
∫ B • ds = μ0ienc
z
Ampere’s law
z
Combine Ampere’s and Maxwell’s law
r r
dΦ E
∫ B • ds = μ0ε 0 dt + μ0ienc
z
B field can be produced by a current and/or a
changing E field
z
z
Wire carrying constant current, dΦE /dt = 0
Charging a capacitor, no current so ienc = 0
Maxwell’s laws
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Basis of all electrical and magnetic phenomena
can be described by 4 equations called
Maxwell’s equations
As fundamental to electromagnetism as
Newton’s law are to mechanics
Einstein showed that Maxwell’s equations work
with special relativity
Maxwell’s equations basis for most equations
studied since beginning of semester and will be
basis for most of what we do the rest of the
semester
Maxwell’s laws
Maxwell’s 4 equations are
z
z
Gauss’ Law
r r q
∫ E • dA =
ε0
Gauss’ Law for magnetism
r r
∫ B • dA = 0
r r
d r r
B • dA
z Faraday’s Law ∫ E • ds = −
∫
dt
z
r r
r r
d
Ampere-Maxwell Law B • ds = ε 0 μ0
E • dA + μ0i
∫
∫
dt
Maxwell’s laws in differential form
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Gauss’ Law
r r ρ
∇•E = ;
ε0
ρ = charge density
r r
Gauss’ Law for magnetism ∇ • B = 0
r
r r
dB
Faraday’s Law ∇ &times; E = − dt
r
r r 1 dE
r
Ampere-Maxwell Law ∇ &times; B =
+ μ0 J
2
c dt
r
1
2
c =
;
J = current density
μ 0ε 0
```