PHY2049: Chapter 30 1

SubjectsInduced emf

Faraday’s law (law #3 of electricity and magnetism)Lenz’s lawMotional emfGenerator

Inductors and Inductance

RL circuits

Magnetic energy

Transformers

Section 30-6 (Induced Electric Field): read the book!

Important concept. Needed in Chapters 32 & 33

PHY2049: Chapter 30 2

InductorsCome in many shapes and sizes. Most common

are small solenoids and toroids. Used in electronic circuits.

PHY2049: Chapter 30 3

Self-induced emfWhat is it?

If a current through a coil of wire changes, B produced by it changes. This causes magnetic flux in the coil to change, leading to induced emf — self-induced emf.

Self-induced emf can destroy a poorly-designed current supply for an electromagnet. Also, never touch the cables carrying the current to an electromagnet. (See a demo later.)

i

B

PHY2049: Chapter 30 4

InductanceMeasure of the strength of self-induced emf with

respect to current change

Definition: Inductance L of a coil with a uniform radius

Why is this a good definition?Faraday’s law: This is actually the general definition of LNegative sign means: self-induced emf opposes change

in current (just Lenz’s law!)Units

Wb/A = T m2/A = henry = HInductors used in electronics: µHLarge electromagnets in labs: H

iΦNL B≡

dtΦdN Bε −= dt

diLLε −=

PHY2049: Chapter 30 5

CHECKPOINT 5

The figure shows the direction of a self-induced emf of a coil. Which of the following can describe the current through the coil?

(a) constant and rightward

(b) constant and leftward

(c) increasing and rightward

(d) increasing and leftward

Lentz’s law

PHY2049: Chapter 30 6

Example

L of a long solenoid

Definition

Field produced by long solenoid

To increase L: increase n (many turns/length), A (large cross section), l (long solenoid)

Two units for µ0

H/m and T m/A

niµB 0=

AlnµL 20=

iΦNL B≡

PHY2049: Chapter 30 7

CHECKPOINT 6

The three inductors are identical, as are all the resistors and batteries. When the switch is closed at t=0, which circuit has the largest current through the battery?

(a) 1(b) 2(c) 3(d) 1 and 2(e) 1 and 3 (1) (2) (3)

If i through L jumps from 0 to a non-zero value at t=0, then di/dt=∞. emf produced by L would be ∞, which is unphysical. di/dt is finite and i=0.

PHY2049: Chapter 30 8

The same three circuits, with identical L, R, and batteries. Long time after the switch is closed, which circuit has the largest current through the battery?

(a) 1

(b) 2

(c) 3

(d) 1 and 2

(e) 1 and 3

(continued)

At t→∞, di/dt=0. So emf produced by L is 0. Inductor is then just a piece of wire.

(1) (2) (3)

PHY2049: Chapter 30 9

Reaching steady state takes timeSelf-induced emf of inductor opposes current change from 0Current takes time to reach full value

Close switch at t=0Initial current: i = 0Initial emf of L: ℇL = ℇ of battery

At t ∞Final current: i = ℇ/RFinal emf of L: ℇL = 0

(since di/dt=0. L behaves just as piece of wire.)

RL Circuit

PHY2049: Chapter 30 10

What happens in between? Use loop rule:E – i R – L di /dt = 0

Solve the differential equation

General solution is(Check and see!)K = −E /R (necessary to make i = 0 at t = 0)

Current i and self-induced emf of L

Compare with q and i in RC circuit

(continued)

LLRi

dtdi ε

=+

)//( RLtKeR

i ε −+=

)//(|| RLtL eiR εεε −=−=( ))//(1 RLte

Ri ε −−=

PHY2049: Chapter 30 11

Current and V vs Time

( ))//(1 RLteR

i ε −−=

self-induced emf of inductor

)//(|| RLteεεL−=

t/(L/R)

( ))//(1 RLteV ε −−=across resistor

PHY2049: Chapter 30 12

Question 8

Consider three circuits wired as shown, containing the same resistance R and battery but different inductance L. Which of the three graphs shows the potential difference VR across the circuit with the smallest L, after the switch is closed?

abc

PHY2049: Chapter 30 13

The three circuits have identical L, R, and batteries. Which circuit has the shortest time constant?

(a) 1

(b) 2

(c) 3

(d) 1 and 2

(e) 1 and 3

Time Constant

To find tau for circuits 2 or 3, use loop rule to write a differential equation. No need to solve it.

(1) (2) (3)

PHY2049: Chapter 30 14

di/dt = –∞ will cause εL = – L di/dt = ∞

Sparks! Can electrocute you ☹

Use a make-before-break switch to provide a current path

Loop rule: – i R – L di /dt = 0

SolveGeneral solution isK=ℇ/R to make i=ℇ/R at t=0

RL Circuit 2 (Disconnect battery)

0=+LRi

dtdi

)//( RLtKei −=)//( RLte

Ri ε −= )//( RLteiRV ε −==

PHY2049: Chapter 30 15

Current and V vs Time

)//( RLteV ε −= ( ))//(1 RLteR

i ε −−=

)//( RLteV ε −=

)//( RLteV ε −=

)//( RLteR

i ε −=

across resistor

t/(L/R)

self-induced emf of inductor

)//(|| RLteεεL−=