The story so far…

dIMagnetic field generated by current element: Biot-Savart

Ampere’s law

dB

r

€

dB =μo

4π

Ids × ˆ r

r2

€

B • ds∫ = μoI

closed path

surface bounded by path

I

Mon. Mar. 31, 2008 Physics 208, Lecture 18 2

Exam 2 results

Grade cutoffs:A: 86AB: 79B: 66BC: 58C: 37D: 23

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Phy208 Exam 2

SCORE

Ave=69

Mon. Mar. 31, 2008 Physics 208, Lecture 18 3

Ampere’s lawSum up component of B around path

Equals current through surface.

Ampere’s law

€

B • ds∫ = μoI

closed path

surface bounded by path

I

€

rB

Component of B along path

Mon. Mar. 31, 2008 Physics 208, Lecture 18 4

“Ampere’s law” in electrostatics

€

rE • d

r s

path

∫ = ?

€

WAB =r F Coulomb • ds

A

B

∫ = qr E • ds

A

B

∫Work done by E-field =

So

is work per unit charge to

bring charge back to where it started.

€

rE • d

r s

path

∫

This is zero.

Mon. Mar. 31, 2008 Physics 208, Lecture 18 5

Gauss’ law in electrostatics

Electric flux through surface charge enclosed

What about magnetic flux?

Mon. Mar. 31, 2008 Physics 208, Lecture 18 6

Magnetic flux Magnetic flux is defined

exactly as electric flux (Component of B surface) x (Area element)

€

ΦB = B • dA∫

zero flux Maximum flux

SI unit of magnetic flux is the Weber ( = 1 T-m2 )

Mon. Mar. 31, 2008 Physics 208, Lecture 18 7

Magnetic flux

What is that magnetic flux through this surface?

A. Positive

B. Negative

C. Zero

Mon. Mar. 31, 2008 Physics 208, Lecture 18 8

Gauss’ law in magnetostatics Net magnetic flux through any closed

surface is always zero:

€

Φmagnetic = 0

No magnetic ‘charge’, so right-hand side=0 for mag.

Basic magnetic element is the dipole

€

Φelectric =Qenclosed

εo

Compare to Gauss’ law for electric field

Mon. Mar. 31, 2008 Physics 208, Lecture 18 9

Comparison with electrostatics

Gauss’ law Ampere’s law

Magnetostatics

Electrostatics

Mon. Mar. 31, 2008 Physics 208, Lecture 18 10

Time-dependent fields Up to this point, have discussed only magnetic

and electric fields constant in time. E-fields arise from charges B-fields arise from moving charges (currents)

Faraday’s discovery

Another source of electric field Time-varying magnetic field creates electric field

Mon. Mar. 31, 2008 Physics 208, Lecture 18 11

Measuring the induced field

A changing magnetic flux produces an EMF around the closed path.

How to measure this? Use a real loop of wire for the closed path.

The EMF corresponds to a current flow:

€

ε=IR

Mon. Mar. 31, 2008 Physics 208, Lecture 18 12

Current but no battery?

Electric currents require a battery (EMF) Faraday:

Time-varying magnetic field creates EMF

Faraday’s law:

EMF around loop = - rate of change of mag. flux

Mon. Mar. 31, 2008 Physics 208, Lecture 18 13

Faraday’s law

€

ε = E • ds∫ = −d

dtΦB = −

d

dtB∫ • dA

Magnetic flux through surface bounded by path

EMF around loop

EMF no longer zero around closed loop

Mon. Mar. 31, 2008 Physics 208, Lecture 18 14

Quick quiz Which of these conducting loops will have currents flowing in them?

Constant I I(t) increases

Constant IConstant v

Constant I

Constant v

Mon. Mar. 31, 2008 Physics 208, Lecture 18 15

Faraday’s law

Faraday’s law Time-varying B-field creates E-field Conductor: E-field creates electric current

Biot-Savart law Electric current creates magnetic field

Result Another magnetic field created

Mon. Mar. 31, 2008 Physics 208, Lecture 18 16

Lenz’s law Induced current produces a magnetic field.

Interacts with bar magnet just as another bar magnet

Lenz’s law Induced current generates a magnetic field

that tries to cancel the change in the flux.

Here flux through loop due to bar magnet is increasing. Induced current produces flux to left.

Force on bar magnet is to left.

Mon. Mar. 31, 2008 Physics 208, Lecture 18 17

Quick quizWhat direction force do I feel due to Lenz’ law when I

push the magnet down?

A. Up

B. Down

C. Left

D. Right

Copper

Strong magnet

Mon. Mar. 31, 2008 Physics 208, Lecture 18 18

Quick Quiz

A conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown.

What is the direction of the induced loop current?

X X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X X

v

x

y

A. CCW

B. CW

C. No induced current

Mon. Mar. 31, 2008 Physics 208, Lecture 18 19

Quick Quiz

•Conducting rectangular loop moves with constant velocity v in the -y direction away from a wire with a constant current I as shown. What is the direction of the induced loop current?

A. CCW

B. CW

C. No induced current

I

v

B-field from wire into page at loopLoop moves to region of smaller B, so flux decreasesInduced loop current opposes this change, so must create a field in same direction as field from wire -> CW current.

Mon. Mar. 31, 2008 Physics 208, Lecture 18 20

L

Motional EMF Conductor moving in uniform magnetic field + / - charges in conductor are moving. Magnetic field exerts force.

€

rv

-

€

r v

€

rF B

Charges pile up at ends

Static equilibrium: E-field generated canceling magnetic force

€

qE = qvB

€

EMF = vBLSolid conductor