1

LECTURE 18

•  More on Self Inductance •  Calculation of Self-Inductance for

Simple Cases •  RL Circuits •  Energy in Magnetic Fields

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Self Inductance •  The magnetic field produced by the

current in the loop shown is proportional to that current. I

•  The flux is also proportional to the current.

• Define the constant of proportionality between flux and current to be the inductance, L

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Self Inductance

•  Archetypal inductor is a long solenoid, just as a pair of parallel plates is the archetypal capacitor

d

A

- - - - - + + + +

r << l l

r N turns d << A

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RL Circuits R

I

"

a

b

L

I •  At t=0, the switch is closed and

the current I starts to flow.

Initially, an inductor acts to oppose changes in current through it. A long time later, it acts like an ordinary connecting wire.

2

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RL Circuits R

I

"

a

b

L

I •  Find the current as a function of

time.

I =εR1− e−Rt /L( ) = ε

R1− e− t /τRL( )

•  What about potential differences?

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RL Circuits ( on)

Current

Max = /R

63% Max at t=L/R

L/R

t

I

2L/R

0 0 1 2 3 4

0

0.5

1

t/RC

Q f( )x

x

/R

VL

0 t

"

0 1 2 3 4

0.5

11

0.0183156

f( )x

40 x

Voltage on L

Max = /R 37% Max at t=L/R

I =εR1− e−Rt /L( )

VL = LdIdt

= εe−Rt /L

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RL Circuits R

I

"

a

b

L

I •  Why does RL increase for

larger L?

•  Why does RL decrease for larger R?

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RL Circuits R

I

"

a

b

L

I After the switch has been in position for a long time, redefined to be t=0, it is moved to position b.

3

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RL Circuits ( off)

0

0 1 2 3 40

0.5

1

t/RC

Q f( )x

x

-"

VL

t

L/R

t

2L/R

I

0

/R

0 1 2 3 4

0.5

11

0.0183156

f( )x

40 x

Current

Max = /R 37% Max at t=L/R

Voltage on L

Max = - 37% Max at t=L/R

I =εRe−Rt /L

VL = LdIdt

= −εe−Rt /L

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Inductor in Series

L2

i i

L1

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Inductor in Parallel

L1 L2

i i1

i2

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Energy in an Inductor •  How much energy is stored in

an inductor when a current is flowing through it?

•  Start with loop rule:

•  PL, the rate at which energy is being stored in the inductor:

•  Integrate this equation to find U, the energy stored in the inductor when the current = I:

•  Multiply this equation by I:

R I

"

a

b

L

I

ε = IR + LdIdt

ε I = I 2R + LIdIdt

PL =dUdt

= LIdIdt

U = dU = LIdI0

I

∫0

U

∫ U =12LI 2

4

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Where is the Energy Stored? •  Claim: (without proof) energy is stored in the magnetic

field itself (just as in the capacitor / electric field case).

•  To calculate this energy density, consider the uniform field generated by a long solenoid:

•  The inductance L is:

•  Energy U:

•  The energy density is found by dividing U by the volume containing the field:

l

r

N turns

B = µ0NlI

L = µ0N 2

lπr2

U =12LI 2 =

12

µ0N 2

lπr2

⎛⎝⎜

⎞⎠⎟I 2 =

12πr2l B

2

µ0

u =Uπr2l

=12B2

µ08/16/12 14

Mutual Inductance

1 22 1 and dI dIM M

dt dtε ε= − = −

A changing current in a coil induces an emf in an adjacent coil. The coupling between the coils is described by mutual inductance M.

M depends only on geometry of the coils (size, shape, number of turns, orientation, separation between the coils).

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Mutual Inductance Applications Superconductors

Resistivity versus temperature for an ordinary metal

Resistivity versus temperature for ‘superconductor’

infinite mobility!

5

1.  Cool it down 2.  Move magnet What will happen?

dtd

emf magΦ−=

Current in the loop will produce its own B to compensate for any changes in magnetic flux.

Magnetic Flux Through a Superconducting Ring

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Superconductors

�

Resistivity ρ = 0 for temperature T < Tc.A type 1 superconductor is a perfect diamagnetic material with χm = −1. B = B app(1+ χm )

Meissner effect (1939)

B field becomes =0 because superconducting currents on the surface of the superconductor create a magnetic field that cancel the applied one for T<Tc

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Magnetic levitation:

Repulsion between the permanent Magnetic field producing the applied field and the magnetic field produced by the currents induced in the superconductor

magnetic levitation of a type 1 superconductor

DEMO

Superconductor: YBa2Cu3O7

Magnet: NdFeB