Physics 122B Electricity and Magnetism

Lecture 23 (Knight 33.8-33.10) Inductance and LR

Circuits

- Martin Savage

Lecture 23 Announcements

- Lecture HW has been posted and is due on

Wednesday at 10 PM. - Midterm examination 3 is this coming

Friday---questions from lecture, tutorial AND Lab.

Generators

The figure shows a coil with N turns rotating

in a magnetic field, with the coil connected to

an external circuit by slip rings that transmit

current independent of rotation. The flux

through the coil is

Therefore, the device produces emf and

current that will vary sinusoidally, alternately

positive and negative. This is called an

alternating current generator, producing what we

call AC voltage.

Example An AC Generator

A coil with area 2.0 m2 rotates in a 0.10 T

magnetic field at a frequency of 60 Hz. How many

turns are needed to generate an AC emf with a

peak voltage of 160 V?

Transformers

When a coil wound around an iron core is

driven by an AC voltage V1cos wt, it produces an

oscillating magnetic field that will induce

an emf V2cos wt in a secondary coil wound on the

same core. This is called a transformer.

The input emf V1 induces a current I1 in the

primary coil that is proportional to 1/N1. The

flux in the iron is proportional to this, and it

induces an emf V2 in the secondary coil that is

proportional to N2. Therefore, V2 V1(N2/N1).

From conservation of energy, assuming no losses

in the core, V1I1 V2I2. Therefore, the

currents in the primary and secondary are related

by the relation I1 I2(N2/N1). A

transformer with N2gtgtN1 is called a step-up

transformer, which boosts the secondary voltage.

A transformer with N2ltltN1 is called a step-down

transformer, and it drops the secondary voltage.

The Tesla Coil

A special case of a step-up transformer is

the Tesla coil. It uses no magnetic material,

but has a very high N2/N1 ratio and uses

high-frequency electrical current to induce very

high voltages and very high frequencies in the

secondary. There is a phenomenon called

the skin effect that causes high frequency AC

currents to reside mainly on the outer surfaces

of conductors. Because of the skin effect, one

does not feel (much) the electrical discharges

from a Tesla coil.

Metal Detectors

Metal detectors like those used at airports

can detect any metal objects, not just magnetic

materials like iron. They operate by induced

currents. A transmitter coil sends high

frequency alternating currents that will induce

current flow in conductors in its field. Because

of Lenzs Law, the induced current opposes the

field from the transmitter, so that net field is

reduced. A receiver coil detects the reduction

in the magnetic fields from the transmitter and

registers the presence of metal.

(Self-) Inductance

We define the inductance L of a coil of wire

producing flux Fm as

The unit of inductance is the henry 1

henry 1 H 1 T m2/A 1 Wb/A

The circuit diagram symbol used to represent

inductance is

Example The inductance of a long solenoid

with N turns of cross sectional area A and length

l is

Example Length of an Inductor

An inductor is made by tightly winding 0.30

mm diameter wire around a 4.0 mm diameter

cylinder. What length cylinder has an

inductance of 10 mH?

Potential Across an Inductor

Potential Across an Inductor (2)

Inductors and Sparks

An inductor responds to an interruption in

current flow by developing a very large potential

(L dI/dt) across its terminals. This potential

is typically dropped across the object (usually a

switch) that is effecting the change in current.

Therefore, breaking a closed circuit with an

inductor often results in a spark across the

switch contacts. Over time, this can erode and

destroy the switch. Spark plugs in

automobiles are driven by inductances that are

electronically interrupted.

Example Large Voltage across an Inductor

A 1.0 A current passes through a 10 mH

inductor coil. What potential difference is

induced across the coil if the current drops to

zero in 5 ms?

( 1.0 A )

Question

The potential at a is higher than the

potential at b. Which of the following

statements about the inductor current I is

consistent with this observation?

- I is from a to b and steady.
- I is from a to b and increasing
- I is from a to b and decreasing
- I is from b to a and steady
- I is from b to a and increasing

Energy in Inductors and Magnetic Fields

A magnetic field stores considerable energy.

Therefore, an inductor, which operates by

creating a magnetic field, stores energy. Lets

consider how much energy UL is stored in an

inductor L carrying current I

A solenoid of length l , area A, and N turns has

L m0N2A/l, so

Energy Density in the Fields

This is the magnetic analog of the energy stored

in an electric field

Magnetic

Electric

The LR Circuit

Example Exponential Decay in an LR Circuit

The switch shown has been in position a for

a long time. It changes to position b at t0.

a. What is the current in the circuit at t 5

ms? b. At what time has the current decayed

to 1 of its original value?

Plumbers RL Analogy

Constriction

Valve

P1

P2

Pump

Flywheel

The plumbers analogy of an RL circuit is a

pump (battery) pumping water in a closed loop of

pipe that includes a valve (switch), a

constriction (resistor), and rotating flywheel

turned by the flow. When the valve starts the

flow, the flywheel begins to spin until a steady

flow is achieved and the pressure difference

across the flywheel (P2-P3) goes to zero.

P3

Pump Battery Valve Switch Constriction

Resistor Flywheel Inductor Pressure

Potential Water Flow Current

Capacitors Early and Late

Initially, when a switch closes there is a

potential difference of 0 across an uncharged

capacitor. After a long time, the capacitor

reaches its maximum charge and there is no

current flow through the capacitor. Therefore,

at t0 the capacitor behaves like a short circuit

(R0), and at t8 the capacitor behaves like at

open circuit (R8). Example

100 V

100 V

12.5 A

100 V

10 A

2.5 A

0 V

at t0

at t8

Circuit

Calculate initial currents. IB 100 V/8 W

12.5 A

Calculate final potentials.

From Lecture 16

Inductors Early and Late

Initially, when a switch closes an inductor

appears to have an infinite resistance and has a

maximum potential drop across it. Ultimately,

the inductor reaches a steady current flow with

no potential drop across it. Therefore, at t0

the inductor behaves like at open circuit (R8),

and at t8 the inductor behaves like a short

circuit (R0). This behavior is opposite that of

a conductor. Example

at t0

at t8

Circuit

Calculate initial potentials.

Calculate final currents.

Question

Rank the order of the time constants in these

circuits.

- t1t2t3 (b) t1ltt2ltt3 (c) t2ltt1ltt3

- (d) t1gtt2gtt3 (e) t2gtt1gtt3

Lecture 23 Announcements

- Lecture HW has been posted and is due on

Wednesday at 10 PM. - Midterm examination 3 is this coming

Friday---questions from lecture, tutorial AND Lab.