Lecture 17-Lecture 17-11 Ampere’s Law in Magnetostatics

The path integral of the dot product of magnetic field and unit vector along a closed loop, Amperian loop, is proportional to the net current encircled by the loop,

0t CC CB dl B dl I

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• Choosing a direction of integration.

• A current is positive if it flows along the RHR normal direction of the Amperian loop, as defined by the direction of integration.

Biot-Savart’s Law can be used to derive another relation: Ampere’s Law

0 1 2( )i i

Lecture 17-Lecture 17-22Magnetization and “Bound” Current in Matter

• Strong externally applied field Bapp aligns the magnetic moments in matter. Magnetization

MV

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dV

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m appmB B ����������������������������

0

mapp appM B M B

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0 (1 )a a pmpp pB MB B ��������������������������������������������������������

00

appappm m

BB K B K

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Magnetic susceptibility

Relative permeability Km

permeability

Lecture 17-Lecture 17-33Hysteresis for a Ferromagnet

Lack of retraceability shown is called hysteresis.

Memory in magnetic disk and tape

Alignment of magnetic domains retained in rock (cf. lodestones)

Area enclosed in hysteresis loop

Energy loss per unit volume

hard magnet: broad hysteresis loop (hard to demagnetize, large energy loss, highe memory)

soft magnet: narrow hysteresis loop (easy to demagnetize,…)

Lecture 17-Lecture 17-44Magnetic Flux

2 cosB BA B nA

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B

S

B ndA ��������������

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iB��������������

1 Wb = 1 T m2

0S

B ndA ��������������Gauss’s Law

for Magnetism

over closed surface

cosB NBA

(N turns)

Lecture 17-Lecture 17-55 Faraday’s Law of Induction

The magnitude of the induced EMF in conducting loop is equal to the rate at which the magnetic flux through the surface spanned by the loop changes with time.

BdΦε

dt where

B SB ndA ��������������

Minus sign indicates the sense of EMF: Lenz’s Law

• Decide on which way n goes

Fixes sign of B

• RHR determines the positive direction for EMF

N

N

Lecture 17-Lecture 17-66

N

1. define the direction of ; can be any of the two normal direction, e.g. point to right

2. determine the sign of Φ. Here Φ>0

3. determine the sign of ∆Φ. Here ∆Φ >0

4. determine the sign of using faraday’s law. Here <0

5. RHR determines the positive direction for EMF • If >0, current follow the direction of the curled

fingers. • If <0, current goes to the opposite direction of

the curled fingers.

n

n

How to use Faraday’s law to determine the induced current direction

Lecture 17-Lecture 17-77 Conducting Loop in a Changing Magnetic Field

Induced EMF has a direction such that it opposes the change in magnetic flux that produced it.

Magnetic moment created by induced currrent I repels the bar magnet.

Magnetic moment created by induced currrent I attracts the bar magnet.

Force on ring is repulsive. Force on ring is attractive.

approaching moving away

Lecture 17-Lecture 17-88 Induced Electric Field from Faraday’s Law

/ε W q• EMF is work done per unit charge:

ncW q E ds

• If work is done on charge q, electric field E must be present:

ncε E ds

Bnc

dΦE ds

dt

Rewrite Faraday’s Law in terms of induced electric field:

This form relates E and B!

• The induced E by magnetic flux changes is non-conservative.

B��������������

• Note that for E fields generated by charges at rest (electrostatics) since this would correspond to the potential difference between a point and itself. => Static E is conservative.

0E ds

Lecture 17-Lecture 17-99

Warm-up quiz

The magnetic field is decreasing, what’s the direction of the induced currents in the closed rectangular loop?

A. Clockwise

B. Counterclockwise

C. No induced currents.

Lecture 17-Lecture 17-1010Faraday’s and Lenz’s Laws

At 1, 3, and 5, B is not changing. So there is no induced emf.

At 2, B is increasing into page. So emf is induced to produce a counterclockwise current.

At 4, B in decreasing into page. So current is clockwise.

Lecture 17-Lecture 17-1111Motional EMF of Sliding Conductor

Lenz’s Law gives direction

Induced EMF:

counter-clockwise

Faraday’s Law

Bd dxBl Blv

dt dt

This EMF induces current IBlv

IR R

Magnetic force FM acts on this I2 2

M

B l vF I lB

R

FM decelerates the bar2 2dv B l v

mdt R

2 2dv B ldt

v mR

2 2

( ) 0B l

tmRv t v e

Lecture 17-Lecture 17-1212 Ways to Change Magnetic Flux

• Changing the magnitude of the field within a conducting loop (or coil).

• Changing the area of the loop (or coil) that lies within the magnetic field.

• Changing the relative orientation of the field and the loop.

motor generator

cosB BA

Lecture 17-Lecture 17-1313 Other Examples of Induction

Switch has been open for some time:

Nothing happening

Switch is just closed:

EMF induced in Coil 2

+ -

Switch is just opened:

EMF is induced again

+-

Switch is just closed:

EMF is induced in coil

+

-

Back emf (counter emf)

Lecture 17-Lecture 17-1414PHYS241 - Quiz A

A current directed toward the top of the page and a rectangular loop of wire lie in the plane of the page. Both are held in place by an external force. If the current I is decreasing, what is the direction of the magnetic force on the left edge of the loop?a. Toward the right

b. Toward the left

c. Toward top of page

d. Toward bottom of page

e. No force acts on it.

I

Lecture 17-Lecture 17-1515PHYS241 - Quiz B

A current directed toward the top of the page and a rectangular loop of wire lie in the plane of the page. If the current I is increasing, what happens to the loop?

a. The loop is pulled toward the top of the page

b. The loop is pulled toward the current

c. A clockwise current is induced in the loop.

d. A counterclockwise current is induced in the loop.

e. Nothing happens to the loop

I

Lecture 17-Lecture 17-1616PHYS241 - Quiz C

A current directed toward the top of the page and a circular loop of wire lie in the plane of the page. If a clockwise current is induced in the loop by the current I, what can you conclude about it?

a. I is increasingb. I is decreasingc. I remains constantd. I is discontinuous e. Nothing can be

said.

I