Physics 1502: Lecture 22Today’s Agenda

• Announcements:– RL - RV - RLC circuits

• Homework 06: due next Wednesday …Homework 06: due next Wednesday …

• Induction / AC current

InductionSelf-Inductance, RL Circuits

X X X X X X X X X

RI

ε

a

b

L

I

long solenoid

Energy and energy density

t

I

0

εRL/R 2L/R

VL

0t

ε

ε on ε off

0

-ε

VL

t

L/R 2L/R

t

I

0

εR

t

q

0

CεRC 2RC

I

0t

εR

Charging Discharging

0

- εR

I

t

RC 2RC

t

q

0

Cε

Mutual Inductance• Suppose you have two coils

with multiple turns close to each other, as shown in this cross-section

• We can define mutual inductance M12 of coil 2 with respect to coil 1 as:

Coil 1 Coil 2

B

N1 N2

It can be shown that :

Inductors in Series• What is the combined (equivalent)

inductance of two inductors in series, as shown ?

a

b

L2

L1

a

b

LeqNote: the induced EMF of two inductors now adds:

Since:

And:

Inductors in parallel• What is the combined (equivalent)

inductance of two inductors in parallel, as shown ?

a

b

L2L1

a

b

LeqNote: the induced EMF between points a and be is the same !

Also, it must be:

We can define:

And finally:

LC Circuits

• Consider the LC and RC series circuits shown:

LCC R

• Suppose that the circuits are formed at t=0 with the

capacitor C charged to a value Q. Claim is that there is a qualitative difference in the time development of the currents produced in these two cases. Why??

• Consider from point of view of energy!

• In the RC circuit, any current developed will cause energy to be dissipated in the resistor.

• In the LC circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor!

RC/LC Circuits

RC:

current decays exponentially

C R

i

Q

-it

0

0 1

+++

- - -

LC

LC:

current oscillates

i

0 t

i

Q+++

- - -

LC Oscillations(qualitative)

LC+ +

- -

LC

LC+ +

- -LC

Energy transfer in a resistanceless, nonradiating LC circuit.

The capacitor has a charge Qmax at t = 0, the instant at which the switch is closed.

The mechanical analog of this circuit is a block–spring system.

LC Oscillations(quantitative)

• What do we need to do to turn our qualitative knowledge into quantitative knowledge?

• What is the frequency of the oscillations (when R=0)?

– (it gets more complicated when R finite…and R is always finite)

LC+ +

- -

LC Oscillations(quantitative)

• Begin with the loop rule:

• Guess solution: (just harmonic oscillator!)

where: • determined from equation

• , Q0 determined from initial conditions • Procedure: differentiate above form for Q and substitute into

loop equation to find .

LC+ +

- -

i

Q

remember:

Review: LC Oscillations

• Guess solution: (just harmonic oscillator!)

where: • determined from equation

• , Q0 determined from initial conditions

LC+ +

- -

i

Q

1

which we could have determinedfrom the mass on a spring result:

The energy in LC circuit conserved !

When the capacitor is fully charged:

When the current is at maximum (Io):

At any time:

The maximum energy stored in the capacitor and in the inductor are the same:

Lecture 22, ACT 1• At t=0 the capacitor has charge Q0; the resulting

oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I . – What is the relation between 0 and 2 , the

frequency of oscillations when the initial charge = 2Q0 ?

(a) 2 = 1/2 0 (b) 2 = 0 (c) 2 = 2 0

1A

LC

+ +

- -Q Q=

t=0

Lecture 22, ACT 1• At t=0 the capacitor has charge Q0; the

resulting oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I .

(a) I = I (b) I = 2 I (c) I = 4 I

• What is the relation between I and I , the maximum current in the circuit when the initial charge = 2Q0 ?

1B

LC

+ +

- -Q Q=

t=0

Summary of E&M• J. C. Maxwell (~1860) summarized all of the work on

electric and magnetic fields into four equations, all of which you now know.

• However, he realized that the equations of electricity & magnetism as then known (and now known by you) have an inconsistency related to the conservation of charge!

I don’t expect you to see that these equations are inconsistent with conservation of charge, but you should see a lack of symmetry here!

Gauss’ Law

Gauss’ LawFor Magnetism

Faraday’s Law

Ampere’s Law

Ampere’s Law is the Culprit!• Gauss’ Law:

• Symmetry: both E and B obey the same kind of equation (the difference is that magnetic charge does not exist!)

• Ampere’s Law and Faraday’s Law:

• If Ampere’s Law were correct, the right hand side of Faraday’s Law should be equal to zero -- since no magnetic current.

• Therefore(?), maybe there is a problem with Ampere’s Law.

• In fact, Maxwell proposes a modification of Ampere’s Law by adding another term (the “displacement” current) to the right hand side of the equation! ie

!

Displacement current

Remember:

E

Iin Iout

changing electric flux

Maxwell’s Displacement Current

• Can we understand why this “displacement current” has the form it does?

• Consider applying Ampere’s Law to the current shown in the diagram.

• If the surface is chosen as 1, 2 or 4, the enclosed current = I• If the surface is chosen as 3, the enclosed current = 0! (ie there is no current between the plates of the capacitor)

Big Idea: The Electric field between the plates changes in time. “displacement current” ID = ε0 (dE/dt) = the real current I in the wire.

circuit

Maxwell’s Equations• These equations describe all of Electricity and

Magnetism.

• They are consistent with modern ideas such as relativity.

• They even describe light