1 Chapter 30. 2 Mutual Inductance Consider a changing current in coil 1 We know that B 1 = 0 i 1 N 1 And if i 1 is changing with time, dB 1 /dt= 0.

1

Chapter 30

2

Mutual Inductance

Consider a changing current in coil 1We know that

B1=0i1N1

And if i1 is changing with time, dB1/dt=0N1 d(i1)/dt

But a changing B-field across coil 2 will initiate an EMF2 such that

EMF2=-N2A2 dB1/dtSince dB1/dt is proportional to di1/dt then the

dt

diMEMF

dt

diEMF

12

12

Where M is the mutual inductance which is based on the sizes of the coils, and the number of turns

3

Mutual-mutual Inductance

dt

diMEMF

dt

diEMF

21

21

But it could be that the changes are happening in coil 2. Then

It turns out that this value of M is identical to the previously discussed M so

2

11

1

2212

21 i

N

i

NMwhere

dt

diMEMFand

dt

diMEMF BB

4

My favorite unit—the henry

The Henry (H) is the unit of inductanceEquivalent to:

1H=1 Wb/A = 1 V*s/A = 1 *s = 1 J/A2

H is large unit; typically we use small units such as mH and H.

5

Self Inductance

But a coil of wire with a changing current can produce an EMF within itself.

This EMF will oppose whatever is causing the changing current

So a coil of wire takes on a special name called the inductor

6

Inductor

Definition of inductance is the magnetic flux per current (L)

For an N-turn solenoid, L is L= N/I N turns= (n turns/length)*(l length)

The near center solution of inductance depends only on geometry

Electrical symbol

Anl

LcenterNear

lAnL

i

NLIf

liAnN

inBwherenlBAN

20

20

20

0

7

Inductor

Electrical symbol

dt

diL

dt

dNEMF

ifand

NLii

NLIf

Again, the EMF acts to oppose the change in current

i (increasing)

VL

High potential

Low potential

acts like a

i (decreasing)

VL

Low potential

High potential

acts like a

8

RL Circuits

Initially, S is open so at t=0, i=0 in the resistor, and the current through the inductor is 0.

Recall that i=dq/dt

B

A

V

S

R

L

9

Switch to A

L

Rt

L

Rt

eL

V

dt

diande

R

Vi

Ansatzdt

diLiRV

Vdt

diLiR

1

0B

A

V

S

R

L

Initially, the inductor acts against the changing current but after a long time, it behaves like a wire

i

H

L

10

Voltage across the resistor and inductor

L

Rt

L

Rt

R eVReR

ViRV 11

Potential across resistor, VR

L

Rt

L

L

Rt

L

VeV

eL

VL

dt

diLV

Potential across capacitor, VC

At t=0, VL=V and VR=0

At t=∞, VL=0 and VR=V

B

A

V

S

R

L

11

L/R—Another time constant

L/R is called the “time constant” of the circuit

L/R has units of time (seconds) and represents the time it takes for the current in the circuit to reach 63% of its maximum value

When L/R=t, then the exponent is -1 or e-1

L=L/R

12

Switch to B

The current is at a steady-state value of i0 at t=0

L

Rt

eiti

dt

diL

dt

dqR

dt

diLiR

0)(

0B

A

V

S

R

L

13

Energy Considerations

)2

1(

2

1 22 mvKErecallLiU

dt

diLi

dt

dUP

dt

diLiVi

ViPanddt

diLV

B

B

Rate at which energy is supplied from battery

Rate at which energy is stored in the magnetic field of the inductor

Energy of the magnetic field, UB

14

Energy Density, u

Consider a solenoid of area A and length, l

22

2

1

21

20

0

2

022

0

20

2

Euand

Bu

inBbutinu

Anl

Lbut

lA

Liu

lA

U

volume

Uu

EB

BB

Energy stored at any point in a magnetic field

Energy stored at any point in a magnetic field

15

L-C Oscillator – The Heart of Everything

CL

LCLC

tQLC

tQ

so

tQdt

qd

itQdt

dq

tQqAnsatz

qLCdt

qd

C

q

dt

qdL

dt

qd

dt

dq

dt

d

dt

di

dt

dqi

C

q

dt

diL

10

1

0cos1

cos

cos

sin

cos:

01

0

0

22

2

22

2

2

2

2

2

2

2

If the capacitor has a total charge, Q

16

Perpetual Motion?

17

Starting Points

Charge qCurrent i

t

The phase angle, , will determine when the maximum occurs w.r.t t=0The curves above show what happens if the current is 0 at t=0

18

Energy considerations

A quick and dirty way to solve for i at any time t in terms of Q & q

At t=0, the total energy in the circuit is the energy stored in the capacitor, Q2/2C

At time t, the energy is shared between the capacitor and inductor (q2/2C)+(1/2 Li2)

Q2/2C= (q2/2C)+(1/2 Li2)221

qQLC

i

19

Oscillators is oscillators is oscillators

20

Give me an “R”!

Consider adding a resistor, R to the circuit

The resistor dissipates the energy. For example, consider a child on a swing. His/her father pushes the child and gets the child swinging. In a perfect system, the child will continue swinging forever.

The resistor provides the same action as if the child let their feet drag on the ground. The amplitude of the child’s swing becomes smaller and smaller until the child stops.

The current in the LRC circuit oscillates with smaller and smaller amplitudes until there is no more current

21

Mathematically

2

22

2

2

2

2

2

4

cos)(

0

0

L

R

where

tQetq

AnsatzC

q

dt

dqR

dt

qdL

dt

qd

dt

di

dt

dqi

iRC

q

dt

diL

L

Rt

If R is small,underdamped

When oscillation stops due to R, critically damped

Very large values of R, overdamped

22

Why didn’t I use a voltage source?

The practical applications of the LC, LR, and LRC circuits depend on using a sinusoidally varying voltage source:An AC voltage source