Chapter 30 – Self Inductance, Inductors & DC Circuits Revisited

Self-Inductance and Inductors •  Self inductance determines the magnetic flux in a circuit due to

the circuit’s own current.

•  Self-induced emf – A circuit’s self-inductance will give rise to an emf if the current is changing with time.

•  Inductor - a circuit device designed to have a particularly high self-inductance relative to the rest of the circuit. It is typically a solenoid.

Every circuit has some self inductance!

�B = LI

E = �d�B

dt= �L

dI

dt)

I~E · ~dl = �L

dI

dt

Example: Estimate the self inductance of a 2-cm long solenoid have 200 windings per cm and a radius of 0.3 cm.

� = µ0nIN⇡R2

L = �/I = µ0nN⇡R2

L = (4⇡ ⇥ 10�7)(2⇥ 104)(4⇥ 102)⇡(0.003)2 = 0.3 mH

CT 33.29

What is the current through the resistor a very long time after the switch is closed?

L = 10H V = 10V

R = 20Ω

A) 0 A B) 0.5 A C) 1 A D) 10 A E) other

Clicker Question

CT 33.28

What is the current through the resistor immediately after the switch is closed ?

L = 10H V = 10V

R = 20Ω

A) 0 B) 0.5 A C) 1 A D) 10 A E) other

Clicker Question

Clicker question What  is  the  voltage  at  the  top  and  the  bo1om  of  the  inductor  immediately  a8er  switch  is  closed?  

 

 

 

1.  Vtop  =  10  V,  Vbo,om=  0  

2.  Vtop  =  0,  Vbo,om  =  0  

3.  Vtop  =  10  V,  Vbo,om=  10  V  

4.  WTF  (huh?)?!  This    seems  wrong.  Seems  to  depend  on  how  I  look  at  it…  

L = 10H V = 10V

R = 20Ω

•  Faraday’s Law of induction for the circuit:

•  Self-induced inductance causes line integral to be non-zero!

•  E-field no longer conservative!

Thinking of circuits solely in terms of voltages (and using Kirchoff’s Loop Law) is problematic since the electric field is no longer conservative (it has curl)

I~E · ~dl = �L

dI

dt L = 10H V = 10V

R = 20Ω

X

i

�Vi 6= 0!

Where is the electric field?

Note: It is not correct to think of a voltage difference existing across the inductor (as many books do)

E downward in battery

•  E-field in battery •  E-field in resistor (if

there is a current) •  That’s it! No E-field

anywhere else!!!

I~E · ~dl = �L

dI

dt

Integral of E-field across resistor + integral of E-field across battery = induced emf.

IR� Eb = �LdI

dt

Solution to the Circuit:

timescale to reach steady state:

� = L/R

I(t) =EbR

[1� exp(�Rt/L)]

IR� Eb = �LdI

dt

Follow-up question

Immediately after the switch is closed, what is the reading of the voltmeter V2?

The switch has bin in position a for a long time. What happens when switch is switched to position B?

I(t) =�Vbat

Rexp(�Rt/L)

IR = �LdI

dt

I~E · ~dl = �L

dI

dt

Why do you get a spark as you unplug your hair dryer while it is running?

Clicker  Ques,on  

When  switch  is  opened,  what  happens  to  the  20-­‐Ohm  bulb?  

a)  bulb  instantly  goes  out  (zero  luminosity)  

b)  luminosity  of  bulb  momentarily  stays  the  same  and  then  fades  

c)  Luminosity  of  bulb  instantly  increases  and  then  fades  

Suppose the two resistors are light-bulbs. The switch has been closed for a long time.

Energetics How much energy is dissipated by the resistor after the switch is in position b?

I(t) =�Vbat

Rexp(�Rt/L)

Where does the energy (converted to Ohmic dissipation) come from?

P = I2(t)R = I20R exp[�2Rt/L]

E =

Z 1

0P (t) dt = I20R

Z 1

0exp[�2Rt/L] dt

E =L

2I20

Energy of Magnetic Field •  The idea is that magnetic fields are a form of energy.

•  Inside the solenoid, there is an energy density uB due to the B field.

•  Always true (general result is based on a much more formal derivation)

1

2

LI2= uB ⇥ volume

uB =B2

2µ0

L = µ0nN⇡R2 = µ0nnV

1

2LI2 =

1

2µ0(nI)

2V =1

2

B2

µ0

(V = ⇡R2d)

V

Solar Flare

The LC Circuit

Suppose switch closed at t = 0 with capacitor initially charged to Q. What is q(t)?

Ld2q

dt2+

q

C= 0

� =r

1LC

I = �dq

dtLooks like harmonic oscillator!

m

d

2x

dt

2+ kx = 0 (mass on spring)

��E · �dl = � q

C= �L

dI

dt

q(t) = Q0 cos(!t)

Energy in an LC circuit oscillates between electric and magnetic

•  similar to mechanical harmonic oscillator (kinetic and potential)

LRC Circuit:

With resistance in the circuit, LC oscillations damp out.

Similar to the damped harmonic oscillator:

underdamped case:

I

I⇥E · d� = VC + IR = �L

dI

dt

Ld2q

dt2+ R

dq

dt+

1C

q = 0

q(t) = q0 exp(�R/(2L)t) cos

r1

LC� R2

4L2t

!

+q

-q

I

I =dq

dt

md2x

dt2+ b

dx

dt+ kx = 0

Back emf in motor circuit Circuits involving motors contain a rotating coil in a B field

The rotating coil will generate an emf due to induction.

This emf causes the current to be smaller than what it would be if it weren’t rotating (Lenz’s law).

For this reason we call it a back emf.