Steve Boddeker's 123 Physics Pages;

Ch23 Magnetic Flux and Faraday's Law of Induction

Michael Faraday purposely set out to determine if Oersted's findings can be true in the opposite.

Faraday experimentally showed that a changing magnetic field produced an electric field. This electromotive force due to the electric field in turn applied a force on the free charged in a conductor, inducing current.

Induced Electromotive Force

We should have this experiment in our lab, at least a variation.Coils of wires around an iron bar…increase # of coils, what happens, does current change, does voltage change???(answer at end of chapter)

But what will you definitely see, when the switch is opened and closed, an induced current on the secondary circuit.

As we have been discussing since first day of class…let’s say the red bands are permanent magnets. As the magnet approaches the coil of wire, the magnetic field approaches.

The stationary coil experiences an increasing then decreasing magnetic field, which induces an electric field in the stationary frame of the coil. This emf that the electric field causes the charges to experience a force.

This can be said, at the most simplistic terms, inducing the charges to move…thus current, or induced electricity.

Magnetic Flux

Φ = BA cosθ(remember cosine is for dot products)

Magnetic flux is used in the calculation of the induced emf.

Units are Webers

Example

At a certain location, the Earth's magnetic field has a magnitude of 7×10−5 T and points in a direction that is 60° below the horizontal. Find the magnitude of the magnetic flux through the top of a desk at this location that measures 50 cm by 80 cm.

Φ = BA cosθ

Φ = 7e-5(.5).8 cos(90-60)

Φ = 2.4e-5 Webers

Faraday’s Law of Induction

· Faraday’s law: An emf is induced when the magnetic flux through a loop changes with time.

emf = -N ΔΦ / Δt

Magnetic field lines “change” when the magnet is moving with respect to the coil

Example

Consider a single-loop coil whose magnetic flux is given

a. Is the magnitude of the induced emf in this coil greater near t=4s or near t=5s

b. At what times in this plot do you expect the induced emf in the coil to have a maximum magnitude?

c. Estimate the induced emf in the coil at times near t=1s and t=2s

(a) At 4 seconds, this is analogous to the vertical component of velocity of a ball at the top of its path

(b) Same as above 1, 3, 5 seconds

(c) emf1sec = -N ΔΦ / Δt

emf = -1 (2 - -2) / (0.65 – 1.35)

emf = -4Wb / -.7sec

emf = 5.7 Volts

emf2sec = changing directions

emf2sec= 0 volts

Lenz’s Law

An induced current always flows in a direction that opposes the change that caused it

(Choose either top or bottom to verify the above sketches while using the right hand rule)

Wire in Magnetic Field

Drop Rod (drilled out holes) on rigid vertical wires, in a large magnetic field.

We have an existing magnetic field point out of the page (toward you). A conducting rod connects the right and left side wires. As it falls the magnetic flux decreases, inducing a current.

Eddy Currents

Induced currents, called eddy currents, oppose direction of motion.

Example

Mechanical Work and Electrical Energy

ΔΦ = B ΔA

ΔΦ = B v lΔt

v = Δx/Δt

v Δt = Δx

v Δt l = Δx l

v Δt l = ΔA

emf = -N  ΔΦ / Δt

emf = -N  B v l Δt / Δt

emf = -N  B v l(assume 1 coil)

emf = B v l

emf = B v l

emf/Δx = B v l /Δx

Electric field = B v

E = B v

P = F v

P = B2 v2L2 / R

F = I L B

F = ( V / R) L B

F = (emf / R) L B

F = (B v l / R) L B

F =B2 v L2 / R

Example

Generators and Motors

emf = N  B v l

emf = N  B vmax l

emf = N  B ω r sin(ωt) l

emf = N  B ω l r sin(ωt)

emf = N  B ω A sin(ωt)

emf = N BA ω sin(ωt)

From 121

v = ω r

vmax = ω r sin (ωt) = ω r

An electric motor is exactly the opposite of a generator—it uses the torque on a current loop to create mechanical energy.

Inductance

Inductance is the proportionality constant that tells us how much emf will be induced for a given rate of change in current:

Defined as

emf = L  ΔI / Δt

Then solve for inductance

L = emf (Δt) / ΔI Given emf = N  ΔΦ / Δt

L = N ΔΦ / ΔI

Units henry 1 V sec/A = 1 H

For Solenoids

L = µo n2 A l

Example

RL Circuits

· When the switch is closed, the current immediately starts to increase. The back emf in the inductor is large, as the current is changing rapidly. As time goes on, the current increases more slowly, and the potential difference across the inductor decreases.

τ = L / R

Example

Energy Stored in a Magnetic Field

L = emf (Δt) / ΔI (from above)

emf = N  ΔΦ / Δt

emf = L  ΔI / Δt

Pave = Iave V

Pave = ½ I LΔI/Δt

Pave = ½ I2 L / T

P = Work / time

Pave = U / T

½I2L/T = U / T

U = ½ L I2; remember from capacitors (U = ½CV2)

This circuit has a battery and an inductor, the current in the inductor is increasing with respect to time, where τ = L / R. So what is the average current while charging? Iave = ½ I

, for solenoids L = µo n2 A l

U = ½ L I2

U = ½ µo n2 A l I2

And we know from Ampere’s law B = µo n I

Thus U = ½ B2 A l /µo

We know the volume inside the solenoid is Vol = A length

So Energy/volume µB = B2 / 2µo

Example

Transformers

I2

=

V1

=

N1

I1

V2

N2

P = I V or V = P / I

I2

=

P1 / I1

=

N1

I1

P2 / I2

N2

But energy is conserved, P1 = P2

1 / I1

=

N1

1 / I2

N2

Example