Chapter 29 – Magnetic Induction

Motional EMF

Consider a conductor in a B-field moving to the right.

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V

In which direction will an electron in the bar experience a magnetic force?

( )BvF ×= qB

A uniform electric field is directed downward such that everywhere the net Lorentz force on a charged particle is zero:

The resulting potential difference between the top and bottom of the bar is given by

•  This potential difference is an example of a motional emf. For this case,

•  The emf is due to motion of a conductor in a B field

�V =

Z~E · ~dl = EL = vBL

E = vB

E = vBL

What if the bar were placed across conducting rails (in red) so that there is a closed loop for the electrons to follow?

In this circuit, what direction is the current?

a) clockwise b) counterclockwise

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V L

The motional emf drives the circuit (rather than a battery)

I =ER

=vBL

R

•  What is the net magnetic force on the rod?

•  Consider an external force that pulls the bar to the right at a constant velocity. What is the rate of work done by this external force (input power)?

•  What is the direction of the magnetic force exerted on the rod (due to the current)?

F = ILB =vBL

RLB =

vB2L2

R

Ohmic dissipation!

Where does the energy supplied by the external force go?

Rate of energy dissipated via Ohmic dissipation:

Notice that the rate of Ohmic dissipation is equal to the rate of work done by the external force (as necessitated by conservation of energy)

Application: Generators

Generators supply a motional emf that arises from rotating a loop within a magnetic field.

The resulting emf is time-varying.

Clicker Question A metal bar with zero net charge begins rotating clockwise at a constant rate in a constant, uniform magnetic field pointing into the page, as shown. After the charges have reached static equilibrium within the bar,

a.  there is a uniform electric field along the bar, directed away from the axis.

b.  there is a uniform electric field along the bar, directed towards the axis.

c.  there is a non-uniform electric field along the bar, directed away from the axis.

d.  there is a non-uniform electric field along the bar, directed towards the axis.

e.  there is no electric field within the bar.

Induction •  Motion of wire in B field induces an

emf (thus a current if circuit is closed). This is motional emf, and the emf is due to the magnetic field.

•  Are there other ways? Can an emf be produced (induced) if the circuit is stationary?

Faraday’s Law of Induction •  A changing magnetic field can induce an emf in the circuit.

•  This is a new law of physics! It can’t be explained or derived from previous physical laws that we have studied.

•  Quantitative description uses the concept of magnetic flux

Magnetic flux

�B =Z

�B · d �A

Faraday’s Law of Induction

•  Valid regardless of the reason for the change in magnetic flux:

•  Motional emf

•  Faraday’s Law of Induction

•  “We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of two different phenomena.” — Richard P. Feynman

|E| =����d�B

dt

���� E = �d�B

dt

Motional EMF:

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V L

E = BLv

CT 33.3

A loop of wire is moving rapidly through a uniform magnetic field as shown. Is a non-zero EMF induced in the loop?

A: Yes, there is B: No, there is not

Example A long straight wire has a decreasing current I(t) flowing down a long straight wire. A current loop of length b and width a is situated such that the nearest end is a distance c from the straight wire.

What is the induced current if the resistance of the loop is R?

Applications: Electric Generators

A coil of wire is spun in a magnetic field. This produces an EMF and also a current; both vary with time. (AC-alternating current)

•  Power plant •  Alternator •  Magnetic breaking •  Portable home generator

Real generators use more than one coil!

http://www.sciencejoywagon.com/physicszone/otherpub/wfendt/generatorengl.htm

CT 33.8b

What can you say about the emf generated by the rotating loop at this moment shown (assume a constant angular velocity)?

A) Maximum B) Zero C) Nonzero, and changing

The EMF produced by an AC generator is:

Vemf = 10,000 Volts and f = ω/2π = 60 Hz (for home outlets). The voltage is converted to 120 V via transformers (next chapter)

An energy source is needed to turn the wire coil. Examples include burning coal or natural gas to produce steam (which drives pistons); falling water; burning gasoline in generator engine.

E = �d�B

dt

E = � d

dt(BA cos(!t)) = !BA sin(!t)

Example An electric generator consists of a 100-turn circular coil 50 cm in diameter. Its rotated at f=60 Hz inside a solenoid of radius 75 cm and winding density n = 50 cm-1. What DC current in the solenoid is needed for the the maximum emf of the generator to be 170 V?

Simularity between motors and generators

Lenz’s Law

Lenz’s law is incorporated in Faraday’s law via the minus sign: A positive emf induces a current such that the resulting induced B field contributes a positive magnetic flux. •  For example: an increasing B-field which causes the flux to

increase in time will induce a negative emf, which will induce a current producing a B field in the opposite direction as the original B field.

E = �d�B

dt

Lenz’s law is consistent with conservation of energy; it prevents situations of perpetual motion.

For the sliding bar, it takes a force to the right to keep the bar moving at a steady speed. Without this force, the rod would slide to stop (even without friction)

Note: Induced current “tries” to cause magnetic flux to not change. It always fails.

Lenz’s Law

CT 33.6

A current-carrying wire is pulled away from a conducting loop. As the wire moves, is there a current induced around the loop?

A: Yes, CW

B: Yes, CCW C: No

Which way does the current flow?

Applications of Induction

Magnetic recording and playback:

•  hard drives, tape, credit cards, answering machines, etc

Varying current in solenoid produced varying magnetic field, which aligns magnetic dipoles in material.

Recording:

Varying B in time induces emf in solenoid which produces varying current and voltage.

Playback:

Applications Cont’d •  Dynamic Microphones (sound waves move diaphragm and

attached coil in magnetic field, inducing current)

•  Transformers and inductors (next chapter)

•  Metal Detectors (discussed with Eddy currents)

•  Electric guitar pickup (strings are ferromagnetic)

Spark Plugs

Applications Cont’d Transcrenial magnetic stimulation (TMS):

•  Varying current in coils produce changing magnetic field, which produces an induced emf in brain, triggers electric activity.

The windings of a long solenoid carrying a current I

Where does the emf in the wire loop come from? What actually drives the current? How does the wire “know” that the B field changed inside the solenoid?

Induced Electric Field

•  An electric field is generated (induced) by the changing B field (regardless of whether a wire is there or not).

•  This induced electric field is what produces the EMF.

E =

I~E · ~dl

Faraday’s Law of Induction

•  Notice that this electric field isn’t conservative!

•  An alternative way to express this relationship is in differential form:

•  We say that the electric field has curl.

I~E · ~dl = �d�B

dtI

~E · ~dl = � d

dt

Z~B · ~dA

r⇥ ~E = �@ ~B

@t

Faraday’s Law of Induction

•  Note similarity to Ampere’s Law!

•  Changing magnetic flux (in producing an electric field) plays the role of the current density (in producing a magnetic field)

��E · �dl = �d�B

dt= � d

dt

��B · �dA

I~B · ~dl = µ0Ienc = µ0

Z~J · ~dA

E(r, t) =R2

2r

dB

dt2⇡rE =

dB

dt⇡R2 (r > R)

What is the electric field outside a long solenoid?

Outside solenoid:

��E · �dl = �d�B

dt= � d

dt

��B · �dA

2⇡rE =dB

dt⇡r2

E =dB

dt

r

2(r < R)

��E · �dl = �d�B

dt= � d

dt

��B · �dA

Inside solenoid:

The Betatron

•  The changing magnetic field induces a circumferential electric field which can accelerate a charged particle.

•  Consider the motion of an electron:

•  The magnetic field also confines the electron to a circular path:

2⇡rE = ⇡r2dBav

dt) E =

r

2

dBav

dt

F = dp/dt = qE =qr

2

dBav

dt) p� p0 =

qr

2�Bav

qvBorb

=dp

t

dt= !p = p

v

r) �B

av

= 2�Borb

��E · �dl = �d�B

dt= � d

dt

��B · �dA

Eddy Currents Eddy currents are induced currents in a large (2D or 3D) chunk of a conductor. These currents occur if when the conductor is either moving through a B field or is in changing B field.

Applications: Roller coaster car breaking, induction stoves, magnetic levitation (trains), metal detectors, etc

Example: Faraday Disk Consider a rotating conducting disk within a uniform magnetic field.

What is the motional emf across the disk (from center to edge)?

E =

Z R

0Br! dr =

1

2BR2!

|dV | = Edr = vB dr = Br! dr