Chapter 22

ElectromagneticInduction

22.4 Faraday’s Law of Electromagnetic Induction

FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION

The average emf induced in a coil of N loops is

tN

ttN

o

o

!

!"#=$$

%

&''(

)

#

"#"#=E

SI Unit of Induced Emf: volt (V)

the minus signreminds us thatthe induced emfwill oppose thechange in Φ LENZ’S LAW

E

Faraday’s law states that an emf is generated if the magnetic fluxchanges for any reason. Since Φ = BA cos φ, any change of B, A, or φwill induce an emf.

(The motional emf-Φrelation we derivedis a special case of this.)

where, Φ = BA cos φ

22.5 Lenz’s Law

LENZ’S LAW

The induced emf resulting from a changing magnetic flux has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes the original flux change.

22.5 Lenz’s Law

LENZ’S LAW

The induced emf resulting from a changing magnetic flux has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes the original flux change.

Reasoning Strategy

1. Determine whether the magnetic flux that penetrates the coilis increasing or decreasing.

2. Find what the direction of the induced magnetic field must beso that it can oppose the change in flux by adding or subtracting from the original field.

3. Use RHR-2 to determine the direction of the induced current.

22.5 Lenz’s Law

Conceptual Example 8 The Emf Produced by a Moving Magnet

A permanent magnet is approaching a loop of wire. The external circuit consistsof a resistance. Find the direction of theinduced current and the polarity of the induced emf.

Since the applied magnetic field in the loop is increasing and pointing to the right, Lenz’s law says an induced current will be created in the loop to try to oppose this changeby creating an induced magnetic field to the left.

22.5 Lenz’s Law

Conceptual Example 9 The Emf Producedby a Moving Copper Ring.

There is a constant horizontal magnetic field directed into the page in the shaded region. The field is zero outside the shaded region. A copper ring is dropped vertically through the region.

For each of the five positions, determine whether an induced current exists and, if so, find its direction.

Is the acceleration of the ring the same as it drops through the five positions?

Electric generators

A battery converts chemical energy into electrical energy to produce currents and voltages in circuits.

An electric generator uses Faraday’s Law to convert mechanical energy into electrical energy.

Most of the electric power in the world is produced by electric generators!

(e.g. the 120 V out of your electrical outlet)

22.7 The Electric Generator

HOW AN ELECTRIC GENERATOR PRODUCES AN EMF

An electric generator has essentially the same configuration as an electricmotor, it is just used differently, i.e.,

motor: I mechanical energy generator: mechanical energy I

22.7 The Electric Generator

Equation for the emf, E, induced in a rotating planar coil

For a coil of N loops , loop area, A (valid for any planar shape),turning at a rotational frequency, f, in a magnetic field, B,

E = NABω sin ωt = E0 sin ωt

ω = 2πfangularfrequency inradians/sec

22.7 The Electric Generator

E = E0 sin ωt , E0 = NABω , maximum emf

emf from generators is sinusoidal in time alternating current (ac)

T = period = 1/f = 2π/ω

Example. A generator with a circular coil of 75 turns of area 3.0 x 10-2 m2

is immersed in a 0.20 T magnetic field and rotated with a frequency of60 Hz. Find the maximum emf which is produced during a cycle.

Solution:The maximum emf for a generator is

E0 = NABω

We know N = 75, A = 3.0 x 10-2 m2, B = 0.20 T and f = 60 Hz .

Since ω = 2πf = 2π(60) = 377 radians/s

E0 = (75)(3.0 x 10-2)(0.20)(377) = 170 V

22.9 Transformers

A transformer is a device for increasing or decreasing an ac voltage.

It works on the principle of Faraday’s Law.

Vp

Vs

Vp = -Np ΔΦ/Δt Vs = -Ns ΔΦ/Δt

p

s

p

s

N

N

V

V=

Dividing Vs by Vp cancelsout ΔΦ/Δt since it is the same for the primary and secondarycoils.

Transformerequation

22.9 Transformers

s

p

s

p

p

s

N

N

V

V

I

I==

A transformer that steps up the voltage simultaneously steps down thecurrent, and a transformer that steps down the voltage steps up the current.

Vs = (Ns/Np) Vpp

s

p

s

N

N

V

V=

Ns > Np Vs > Vp step-up transformerNs < Np Vs < Vp step-down transformer

From conservation of energy, the power must be conserved,

Pp = Ps IpVp = IsVs

Example. A certain transformer has 30 turns in is primary coil and500 turns in its secondary coil. If a 12 V ac source is attached to theprimary and 4.0 A is flowing in it, find the voltage and current in thesecondary coil.

Solution. Use the transformer equation

Vs = (Ns/Np) Vp = (500/30)(12) = 200 V step-up voltage

Is = (Np/Ns) Ip = (30/500)(4) = 0.24 A step-down current

Is the power delivered to the secondary coil equal to the power sentto the primary coil?

Pp = IpVp = (4)(12) = 48 W , Ps = IsVs = (0.24)(200) = 48 W

s

p

s

p

p

s

N

N

V

V

I

I==

16.1 The Nature of Waves

Since we will be studying electromagnetic waves, let’s review some general features of waves:

1. A wave is a traveling disturbance.

2. A wave carries energy from place to place.

16.1 The Nature of Waves

Longitudinal Wave - the “disturbance” caused by the wavemoves along the direction that the wave propagates,e.g., sound waves, “compressed slinky waves”……

16.1 The Nature of Waves

Transverse Wave - the “disturbance” caused by the wave movesperpendicular to the direction that the wave propagates,e.g., water waves, “shaken slinky waves”, electromagnetic waves….

16.2 Periodic Waves

Periodic waves consist of cycles or patterns that are produced over and over again by the source.

In the figures, every segment of the slinky vibrates in a simple harmonicmotion, provided the end of the slinky is moved in a simple harmonicmotion.

16.2 Periodic Waves

In the drawing, one cycle is shaded in color.

The amplitude A is the maximum excursion of a particle of the medium fromthe particles undisturbed position.

The wavelength is the horizontal length of one cycle of the wave.

The period is the time required for one complete cycle.

The frequency is the number of cycles per time. It is related to the period and has units of Hz, or s-1.

Tf

1=

16.2 Periodic Waves

!!

fT

v ==Since velocity isdistance/time

The propagation velocity of a periodic wave is related to its frequency andwavelength. Consider the motion of a long train as a periodic wave whichrepeats itself with the passing of each identical car: