Walker, Chapter 23Magnetic Flux and Faraday’s

Law of Induction

Michael Faraday, portrait by Thomas Phillips

2

Induced EMF (Voltage) from changing Magnetic Flux

Electric currents produce magnetic fields.19th century puzzle, can magnetic fields produce currents?A static magnet will produce no current in a stationary coilFaraday: If the magnetic field changes, or if the magnet and

coil are in relative motion, there will be an induced EMF (and therefore current) in the coil.

Key Concept: The magnetic flux through the coil must change, this will induce an EMF in the coil, which produces a current I = / R in the coil.

Such a current is said to be induced by the varying B-field.

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Magnetic Flux

For a “loop” of wire (not necessarily circular) with area A, in an external magnetic field B, the magnetic flux is:

cosBAAB

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Walker problem # 2A uniform magnetic field of 0.0250 T points vertically upward. Find

the magnitude of the magnetic flux through each of the five sides of the open-topped rectangular box shown in the figure, given that the dimensions of the box are L = 31.5 cm, W = 12.0 cm, and H = 10.0 cm.

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Question: Magnetic Field from loop

• Current I flows around the loop as shown, driven by the battery. Inside the loop the magnetic field generated by current I is:

1. Out of screen2. Into screen

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Magnetic flux generated by current in loop.

• In this case, we choose clockwise to be positive current.

• In the plane of this picture, the magnetic field generated by this current is into the picture inside the loop, and out of the picture outside of the loop.

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Examples of Induced Current

Any change of current in primary induces a current in secondary.The secondary shows a deflection only during the instant just after

the switch is closed or opened.

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Induction by Relative Motion• When a permanent magnet

moves relative to a coil, the magnetic flux through the coil changes, inducing an EMF in the coil.

• In a) the flux is increasing• In c) the flux is decreasing

in magnitude.• In a) and c) the induced

current has opposite sign.

v

v

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Faraday’s Law of InductionLenz’s Law

Faraday’s Law: The instantaneous EMF in a circuit (w/ N loops) equals the rate of change of magnetic flux through the circuit:

Nt

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Question: Changing Magnetic Flux• This is a plot of magnetic flux (Wb=T·m2) through a coil.

• For which interval (a, b, or c) is the change in flux 0?

a

b

c

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Walker, Problem 23-9• A 0.25 T magnetic field is perpendicular to a circular loop

of wire with 50 turns and a radius 15 cm.• The magnetic field is reduced to zero in 0.12 s.• What is the magnitude of the induced EMF? (answer 7.35 V)

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Inductance

• Consider a solenoid of length l with N windings and radius r (Area A=r2).

• A current I produces a magnetic field in the solenoid of B = N I / l

• This produces a total flux through each winding of = A B = ( N A / l ) I – Define the Inductance L = ( N2 A / l ) (not a length!!)

• If the current in the solenoid changes, there will be an induced EMF

• = t = ( N2 A / l ) (I/t)• = L (I/t)

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Inductance

Walker Problem #42Determine the inductance of a solenoid with 650 turns in a length

of 25 cm. The circular cross section of the solenoid has a radius of 4.0 cm. [answer: 10.7 mH]

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Example of Inductance

• Consider Solenoid from previous example,

– Radius r = 5cm, length l = 10 cm, 3500 windings.

• L = ( N2 Area / l ) = ( N2 r2 / l )

•

HA

mT

m

m

A

mTL

21.121.1

10.0

05.03500104

2

227

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Transformers• Microscopic currents in soft iron yoke amplify magnetic

field produced by coils, and contain all magnetic field lines to “magnetic circuit” formed by iron.

• Both primary and secondary circuits experience the same magnetic flux for each winding, but have different number of windings.

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Flux in Transformer = flux through a single winding or either coil.

(produced by currents in either or both circuits)

EMF(primary) = NP /t

EMF(secondary) = NS /t

P / S = NP / NS Energy conservation requires

P IP = S IS

P / S = NS / NP

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Walker problem #61A step-up transformer has 20 turns on the primary coil and 500 turns

on the secondary coil. If this transformer is to produce an output of 4600 V with a 12 mA current, what input current and voltage are needed? [answer: 0.3 A, 184 V]