Physics 1402: Lecture 21Today’s Agenda

• Announcements:– Induction, RL circuits

• Homework 06: due next MondayHomework 06: due next Monday

• Induction / AC current

Faraday's Law

dS

B Bv

BN S

v

BS N

InductionSelf-Inductance, RL Circuits

L/R

V

t0

L

X X X X X X X X X

RI

a

b

L

I

Recap from the last Chapter:

• Time dependent flux is generated by change in magnetic field strength due motion of the magnet

• Note: changing magnetic field can also be produced by time varying current in a nearby loop

Faraday's Law of Induction

v

BN S

v

BS N

BdI/dt

Can time varying current ina conductor induce EMF inin that same conductor ?

Self-Inductance• The inductance of an inductor ( a set of coils in some

geometry ..eg solenoid, toroid) then, like a capacitor, can be calculated from its geometry alone if the device is constructed from conductors and air.

• If extra material (eg iron core) is added, then we need to add some knowledge of materials as we did for capacitors (dielectrics) and resistors (resistivity)

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

r <<

l

l

r

N turns

SI UNITS for L : Henry

d

A

- - - - -

+ + + +

Calculationl

r

N turns

• Long Solenoid:

N turns total, radius r, Length l

For a single turn,

The total flux through solenoid is given by:

Inductance of solenoid can then be calculated as:

This (as for R and C) depends only on geometry (material)

RL Circuits

• At t=0, the switch is closed and the current I starts to flow.

• Loop rule:

Note that this eqn is identical in form to that for the RC circuit with the following substitutions:

RI

a

b

L

I

RC: RCRL:

Lecture 21, ACT 1• At t=0 the switch is thrown from position b to

position a in the circuit shown:

– What is the value of the current I a long time after the switch is thrown?

(a) I = 0 (b) I = / 2R (c) I = 2 / R

a

b

R

L

II

R

(a) I = 0 (b) I = / 2R (c) I = 2 / R

1A

• What is the value of the current I immediately after the switch is thrown?

1B

RL Circuits

• To find the current I as a fct of time t, we need to choose an exponential solution which satisfies the boundary condition:

• We therefore write:

• The voltage drop across the inductor is given by:

τRL = LR

R

a

b

L

I I

RL Circuit ( on)

t

I

0

RL/R 2L/R

VL

0t

Current

Max = R

63% Max at t=L/R

Voltage on L

Max = /R

37% Max at t=L/R

RL Circuits• After the switch has been in

position a for a long time, redefined to be t=0, it is moved to position b.

• Loop rule:

• The appropriate initial condition is:

• The solution then must have the form:

R

a

b

L

I I

RL Circuit ( off)

0

-

VL

t

L/R 2L/R

t

I

0

R Current

Max = R

37% Max at t=L/R

Voltage on L

Max = -

37% Max at t=L/R

t

I

0

RL/R 2L/R

VL

0t

on off

0

-

VL

t

L/R 2L/R

t

I

0

R

Review: RC Circuits

(Time-varying currents)• Discharge capacitor:

C initially charged with Q=C

Connect switch to b at t=0.

Calculate current and charge as function of time.

• Convert to differential equation for q:

C

a

b+ +

- -

R

I I

• Loop theorem

Review: RC Circuits

(Time-varying currents)

• Trial solution:

q = Ce- /t RC

• Check that it is a solution:

!

Note that this “guess” incorporates the

boundary conditions:

C

a

b+ +

- -

R

I I• Discharge capacitor:

Review: RC Circuits

(Time-varying currents)

• Current is found from differentiation:

Conclusion:

• Capacitor discharges exponentially with time constant τ= RC

• Current decays from initial max value (= -/R) with same time constant

• Discharge capacitor:

C

a

b+

- -

R+

I I

q = Ce- /t RC

0

-R

I

t

RC 2RC

Discharging Capacitor

t

q

0

C

Current

Max = -/R

37% Max at t=RC

Charge on C

Max = C

37% Max at t=RC

t

q

0

CRC 2RC

I

0t

R

Charging Discharging

0

- R

I

t

RC 2RC

t

q

0

C

Energy of an Inductor• How much energy is stored in

an inductor when a current is flowing through it?

• Start with loop rule:

• From this equation, we can identify PL, the rate at which energy is being stored in the inductor:

• We can integrate this equation to find an expression for U, the energy stored in the inductor when the current = I:

R

a

b

L

I I

• Multiply this equation by I:

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:

l

r

N turns

• The inductance L is:

• We can turn this into an energy density by dividing by the volume containing the field:

• Energy U:

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

LC Oscillations(quantitative)

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

• What is the frequency of the oscillations?

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:

Lecture 21, ACT 2• 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 21, ACT 2• 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