Lecture 21: FRI 06 MAR Lecture 21: FRI 06 MAR Magnetic fields Magnetic fields

Ch.28.8-10

Physics 2102

Jonathan Dowling

28.8 Force on Current in Wire

28.9 Torque on Current Loop

28.10 Magnetic Dipole Moment

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Magnetic Force on a Wire.Magnetic Force on a Wire.

L

BLiFrrr

×= BLiFrrr

×= BLdiFdrrr

×= BLdiFdrrr

×=

. dLr

Br

dFr

i

φ

If we assume the more general case for which the

magnetic field forms an angle with the wire

the magnetic force equation can be written

B φ

Magnetic Force on a Straight Wire in a Uniform Magnetic Field

r

in vector

form as . Here is a vector whose

magnitude is equal to the wire length and

has a direction that coincides with that of the current.

The magnetic force magnitude is s

B

B

F iL B L

L

F iLB

= ×

=

r r r r

in .

In this case we divide the wire into elements of

length , which can be considered as straight.

The magnetic force

dL

φ

Magnetic Force on a Wire of Arbitrary Shape

Placed in a Nonuniform Magnetic Field

on each element is

. The net magnetic force on the

wire is given by the integral .

B

B

dF idL B

F i dL B

×

= ×∫=

r r r

r r r

BF iL B= ×r r r

BdF idL B×=r r r

BF i dL B= ×∫r r r

(28-12)

Torque on a Current Loop: Principle behind electric motors.

Net force on current loop = 0

iaBFF == 31

)sin(1 θFF =⊥

)sin(θτ iabBbFTorque === ⊥

For a coil with N turns,τ = N I A B sinθ, where A is the area of coil

Rectangular coil: A=ab, current = i

But: Net torque is NOT zero!

nNiA ˆ)(=μr n̂,μr

Magnetic Dipole MomentMagnetic Dipole Moment

N = number of turns in coilA=area of coil.

We just showed: τ = NiABsinθRight hand rule:

curl fingers in direction of current;

thumb points along μDefine: magnetic dipole moment μ

Brrr

×=μτ

As in the case of electric dipoles, magnetic dipoles tend to align with the magnetic field.

Electric vs. Magnetic DipolesElectric vs. Magnetic Dipoles

€

UE = −r p ⋅

r E

-Q

θQE

QE

+Q

p=Qa

€

rτ B =r μ ×

r B

nNiA ˆ)(=μr

€

rτ E =r p ×

r E

€

UB = −r μ ⋅

r B

DC Electric Motor