chapter 30 inductance - utk department of physics and ... · pdf filechapter 30 inductance....

Fall 2008Physics 231 Lecture 10-1

Chapter 30Inductance


Magnetic Effects

As we have seen previously, changes in the magnetic flux due to one circuit can effect what goes on in other circuits

The changing magnetic flux induces an emf in the second circuit


Mutual InductanceSuppose that we have two coils,Coil 1 with N1 turns and Coil 2 with N2 turns

Coil 1 has a current i1 which produces a magnetic flux, ΦΒ2 , going through one turn of Coil 2

If i1 changes, then the flux changes and an emf is induced in Coil 2 which is given by

dtd

N B222

Φ−=ε


Mutual InductanceThe flux through the second coil is proportional to the current in the first coil

12122 iMN B =Φ

where M21 is called the mutual inductance

Taking the time derivative of this we get

dtdiM

dtdN B 1

212

2 =Φ

dtdiM 1

212 −=εor


Mutual InductanceIf we were to start with the second coil having a varying current, we would end up with a similar equation with an M12

MMM == 1221We would find that

The two mutual inductances are the same because the mutual inductance is a geometrical property of the arrangement of the two coils

To measure the value of the mutual inductance you can use either

dtdIM 1

2 −=ε ordt

dIM 21 −=ε


Units of Inductance

2AmpJ1

AmpsecV1 Henry 1 =

⋅=


Self InductanceSuppose that we have a coil having N turns carrying a current I

That means that there is a magnetic flux through the coil

This flux can also be written as being proportional to the current

ILN B =Φ

with L being the self inductance having the same units as the mutual inductance


Self InductanceIf the current changes, then the magnetic flux through the coil will also change, giving rise to an induced emf in the coil

This induced emf will be such as to oppose the change in the current with its value given by

dtdIL−=ε

If the current I is increasing, then

If the current I is decreasing, then


Self InductanceThere are circuit elements that behave in this manner and they are called inductors and they are used to oppose any change in the current in the circuit

As to how they actually affect a circuit’s behavior will be discussed shortly


What Haven’t We Talked AboutThere is one topic that we have not mentioned with

respect to magnetic fields

Just as with the electric field, the magnetic field has energy stored in it

We will derive the general relation from a special case


Magnetic Field EnergyWhen a current is being established in a circuit, work has to bedone

If the current is i at a given instant and its rate of change is given by di/dt then the power being supplied by theexternal source is given by

dtdiiLiVP L ==

The energy supplied is given by PdtdU =

The total energy stored in the inductor is then

2

0 21 ILdiiLU

I== ∫


Magnetic Field EnergyThis energy that is stored in the magnetic field is available to act as source of emf in case the current starts to decrease

We will just present the result for the energy density of the magnetic field

0

2

21

µBuB =

This can then be compared to the energy density of an electric field

202

1 EuE ε=


R-L CircuitWe are given the following circuit

and we then close S1 andleave S2 open

It will take some finite amount of time for the circuit to reachits maximum current which is given by RI ε=

Kirchoff’s Law for potential drops still holds


R-L CircuitSuppose that at some time t the current is i

The voltage drop across the resistor is given by RiVab =The magnitude of the voltage drop across the inductor is given by

dtdiLVbc =

The sense of this voltage drop is that point b is at a higher potential than point c so that it adds in as a negative quantity

0=−−dtdiLRiε


R-L Circuiti

LR

Ldtdi

−=εWe take this last equation and

solve for di/dt

Ldtdi

initial

ε=⎟

⎠⎞

⎜⎝⎛Notice that at t = 0 when I = 0 we

have that

Also that when the current is no longer changing, di/dt = 0, that the current is given by R

I ε=

as expected

But what about the behavior between t = 0 and t = ∞


R-L CircuitWe rearrange the original equation and then integrate

( ) ∫∫ −=−

tidt

LR

Ridi

00

'/''

ε( ) dtLR

Ridi

−=− /ε

( )( )tLReR

i /1 −−=εThe solution for this is

Which looks like


R-L CircuitAs we had with the R-C Circuit, there is a time constantassociated with R-L Circuits

RL

=τ

Initially the power supplied by the emf goes into dissipative heating in the resistor and energy stored in the magnetic field

dtdiiLRii += 2ε

After a long time has elapsed, the energy supplied by the emf goes strictly into dissipative heating in the resistor


R-L Circuit

We now quickly open S1and close S2

The current does not immediatelygo to zero

The inductor will try to keep the current, in the same direction, at its initial value to maintain the magnetic flux through it


R-L CircuitApplying Kirchoff’s Law to the bottom loop we get

0=−−dtdiLiR

dtLR

idi

−=Rearranging this we have

and then integrating this

∫∫ −=t

I

dtLR

idi

0

0'

''

0

( )tLReII /0

−=

where I0 is the current at t = 0


R-L Circuit

This is a decaying exponentialwhich looks like

The energy that was stored in the inductor will be dissipated in the resistor


L-C CircuitSuppose that we are now given a fully charged capacitor and an inductor that are hooked together in a circuit

Since the capacitor is fully charged there is a potential difference across it given by Vc = Q / C

The capacitor will begin to discharge as soon as the switch is closed


L-C CircuitWe apply Kirchoff’s Law to this circuit

0=−−Cq

dtdiL

dtdqi =Remembering that

2

2

dtqd

dtdi

=We then have that

012

2=+ q

LCdtqd

The circuit equation then becomes


L-C CircuitThis equation is the same as that for the Simple Harmonic Oscillator and the solution will be similar

)cos(0 ϕω += tQq

The system oscillates with angular frequency

LC1

=ω

ϕ is a phase angle determined from initial conditions

)sin(0 ϕωω +−== tQdtdqiThe current is given by


L-C Circuit

Both the charge on the capacitor and the current in the circuit are oscillatory

The maximum charge and the maximum current occur π / (2ω) seconds apart

For an ideal situation, this circuit will oscillate forever


L-C Circuit


L-C CircuitJust as both the charge on the capacitor and the current through the inductor oscillate with time, so does the energy that is contained in the electric field of the capacitor and themagnetic field of the inductor

Even though the energy content of the electric and magnetic fields are varying with time, the sum of the two at any given time is a constant

BETotal UUU +=


L-R-C Circuit

Instead of just having an L-C circuit with no resistance, what happens when there is a resistance R in the circuit

Again let us start with the capacitor fully charged with a charge Q0 on it

The switch is now closed


L-R-C CircuitThe circuit now looks like

The capacitor will start to discharge and a current will start to flow

We apply Kirchoff’s Law to this circuit and get

0=−−−Cq

dtdiLiR

dtdqi =And remembering that we get

012

2=++ q

LCdtdq

LR

dtqd


L-R-C CircuitThe solution to this second order differential equation is similar to that of the damped harmonic oscillator

The are three different solutions

Underdamped

Critically Damped

Overdamped

Which solution we have is dependent upon the relative values of R2 and 4L/C


L-R-C CircuitUnderdamped:

CLR 42 <

The solution to the second differential equation is then

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎠⎞

⎜⎝⎛−

φtL

RLC

eQqt

LR

2

22

0 41cos

The system still oscillates but with decreasing amplitude, which is represented by the decaying exponential

This solution looks like

This decaying amplitude is often referred to as the envelope


L-R-C Circuit

CLR 42 =Critically Damped:

Here the solution is given by

tL

R

etL

RQq⎟⎠⎞

⎜⎝⎛−

⎟⎠⎞

⎜⎝⎛ += 2

0 21


This is the situation when the system most quickly reachesq = 0


L-R-C Circuit

CLR 42 >Overdamped:

Here the solution has the form

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛+= −⎟

⎠⎞

⎜⎝⎛−

ttt

LR

eLR

eLR

eQq ''20'

21'

212

ωω

ωω


LCLR 14

' 2

2−=ωwith


L-R-C Circuit

The solutions that have been developed for this L-R-C circuit are only good for the initial conditions at t = 0 that q = Q0 and that i = 0

chapter 30 inductance - utk department of physics and ... · pdf filechapter 30 inductance....

Documents