Chapter 31

Faraday’s Law

Faraday’s Experiment• A primary coil is connected to a battery and a

secondary coil is connected to an ammeter

• The purpose of the secondary circuit is to detect current that might be produced by a (changing) magnetic field

• When there is a steady current in the primary circuit, the ammeter reads zero

Faraday’s Experiment• When the switch is opened, the ammeter reads a

current and then returns to zero

• When the switch is closed, the ammeter reads a current in the opposite direction and then returns to zero

• An induced emf is produced in the secondary circuit by the changing magnetic field

• When a magnet moves toward a loop of wire, the ammeter shows the presence of a current

• When the magnet moves away from the loop, the ammeter shows a current in the opposite direction

• When the magnet is held stationary, there is no current

• If the loop is moved instead of the magnet, a current is also detected

Electromagnetic Induction

Electromagnetic Induction

• A current is set up in the circuit as long as there is relative motion between the magnet and the loop

• The current is called an induced current because is it produced by an induced emf

Faraday’s Law and Electromagnetic Induction

• Faraday’s law of induction: the instantaneous emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit

• If the circuit consists of N loops, all of the same area, and if B is the flux through one loop, an emf is induced in every loop and Faraday’s law becomes

dt

dN B

Faraday’s Law and Lenz’ Law• The negative sign in Faraday’s Law is included to

indicate the polarity of the induced emf, which is found by Lenz’ Law:

• The current caused by the induced emf travels in the direction that creates a magnetic field with flux opposing the change in the original flux through the circuit

dt

dN B

Heinrich FriedrichEmil Lenz

1804 – 1865

Faraday’s Law and Lenz’ Law• Example:

• The magnetic field, B, becomes smaller with time and this reduces the flux

• The induced current will produce an induced field, Bind, in the same direction as the original field

dt

dN B

Faraday’s Law and Lenz’ Law• Example:

• Assume a loop enclosing an area A lies in a uniform magnetic field

• Since ΦB = B A cos θ, the change in the flux, ΔΦB, can be produced by a change in B, A or θ

dt

dN B

Chapter 31Problem 17

A coil formed by wrapping 50 turns of wire in the shape of a square is positioned in a magnetic field so that the normal to the plane of the coil makes an angle of 30.0° with the direction of the field. When the magnetic field is increased uniformly from 200 μT to 600 μT in 0.400 s, an emf of magnitude 80.0 mV is induced in the coil. What is the total length of the wire?

Motional emf• A straight conductor of length ℓ

moves perpendicularly with constant velocity through a uniform field

• The electrons in the conductor experience a magnetic force

FB = q v B• The electrons tend to move to the

lower end of the conductor

• As the negative charges accumulate at the base, a net positive charge exists at the upper end of the conductor

Motional emf• As a result of this charge separation,

an electric field is produced in the conductor

• Charges build up at the ends of the conductor until the downward magnetic force is balanced by the upward electric force

FE = q E = q v B; E = v B;

• There is a potential difference between the upper and lower ends of the conductor

Motional emf• The potential difference between the

ends of the conductor (the upper end is at a higher potential than the lower end):

ΔV = E ℓ = B ℓ v• A potential difference is maintained

across the conductor as long as there is motion through the field

• If the motion is reversed, the polarity of the potential difference is also reversed

Motional emf in a Circuit• As the bar (with zero resistance) is

pulled to the right with a constant velocity under the influence of an applied force, the free charges experience a magnetic force along the length of the bar

• This force sets up an induced current because the charges are free to move in the closed path

• The changing magnetic flux through the loop and the corresponding induced emf in the bar result from the change in area of the loop

dt

d B

dt

Blxd )(

dt

dxBl Blv

Motional emf in a Circuit• The induced, motional emf, acts like a

battery in the circuit

• As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases

• The induced current must be in a direction such that it opposes the change in the external magnetic flux (Lenz’ Law)

B vB v and I

R

Motional emf in a Circuit• The flux due to the external field is

increasing into the page

• The flux due to the induced current must be out of the page

• Therefore the current must be counterclockwise when the bar moves to the right

• If the bar is moving toward the left, the magnetic flux through the loop is decreasing with time – the induced current must be clockwise to produce its own flux into the page

Motional emf in a Circuit• The applied force does work on the

conducting bar, thus moving the charges through a magnetic field and establishing a current

• The change in energy of the system during some time interval must be equal to the transfer of energy into the system by work

• The power input is equal to the rate at which energy is delivered to the resistor

2

app Iε

F v B vR

Chapter 31Problem 24

A conducting rod of length ℓ moves on two horizontal frictionless rails. A constant force of magnitude 1.00 N moves the bar at a uniform speed of 2.00 m/s through a magnetic field that is directed into the page. (a) What is the current in an 8.00-Ω resistor R? (b) What is the rate of energy dissipation in the resistor? (c) What is the mechanical power delivered by the constant force?

Induced emf and Electric Fields• An electric field is created in the conductor as a result

of the changing magnetic flux

• Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space (this induced electric field is nonconservative, unlike the electric field produced by stationary charges)

• The emf for any closed path can be expressed as the line integral

• Faraday’s law can be written in a general form:

dt

dsdE B

sdE

Lenz’ Law – Moving Magnet Example• As the bar magnet is moved to the right toward a

stationary loop of wire, the magnetic flux increases with time

• The induced current produces a flux to the left, so the current is in the direction shown

• When applying Lenz’ Law, there are two magnetic fields to consider: changing external and induced

Lenz’ Law – Rotating Loop Example• Assume a loop with N turns, all of the

same area rotating in a magnetic field

• The flux through the loop at any time t is B = BAcos = BAcost

• The induced emf in the loop is

• This is sinusoidal, with max = NABsin

Bdε N

dtNABω ωt

AC Generators

• Alternating Current (AC) generators convert mechanical energy to electrical energy

• Consist of a wire loop rotated by some external means (falling water, heat by burning coal to produce steam, etc.)

• As the loop rotates, the magnetic flux through it changes with time inducing an emf and a current in the external circuit

AC Generators

• The ends of the loop are connected to slip rings that rotate with the loop; connections to the external circuit are made by stationary brushes in contact with the slip rings

• The emf generated by the rotating loop:

sin

Bdε N

dtNABω ωt

DC Generators

• Components are essentially the same as that of an ac generator

• The major difference is the contacts to the rotating loop are made by a split ring, or commutator

• The output voltage always has the same polarity

• The current is a pulsing current

DC Generators

• To produce a steady current, many loops and commutators around the axis of rotation are used

• The multiple outputs are superimposed and the output is almost free of fluctuations

Motors• Motors are devices that convert electrical energy into

mechanical energy (generators run in reverse)

• A motor can perform useful mechanical work when a shaft connected to its rotating coil is attached to some external device

• As the coil begins to rotate, the induced back emf opposes the applied voltage and the current in the coil is reduced

• The induced emf explains why the power requirements for starting a motor and for running it are greater for heavy loads than for light ones

Answers to Even Numbered Problems

Chapter 31:

Problem 28

(a) to the right(b) out of the plane of the paper(c) to the right(d) into the paper