Physics 2102 Physics 2102 Lecture 14Lecture 14

Ch28: Magnetic Forces on Current WiresCh28: Magnetic Forces on Current Wires

Physics 2102

Jonathan Dowling

Star Quake on a

Magnetar!“I’ll be back….

Aurora Borealis

Crossed Fields E vs. BCrossed Fields E vs. B

FE=qE

BEy

q

FB=vqBFE

v

L

FE=ma  =>  y=qEL2/(2mv2)

FB=FE  => y=0  => v=E/B

Magnetic force on a wire.Magnetic force on a wire.

dvL

itiq ==

LBvqF d

rrr×=

BLiBqLi

qFrrr

rr

×=×=

BLiFrrr

×=

BLdiFdrrr

×=

Note: If wire is not straight,compute force on differential elements and integrate:

ExampleExample

iLBFF == 31

θiBRdiBdLdF ==

By symmetry, F2 will only have a vertical component,

iBRdiBRdFF 2)sin()sin(00

2 ∫∫ ===ππ

θθθ

)(22321total RLiBiLBiRBiLBFFFF +=++=++=

Notice that the force is the same as that for a straight wire,

L LR R

and this would be true nomatter what the shape of the central segment!.

Wire with current i.Magnetic field out of page.What is net force on wire?

Example 4: The Rail GunExample 4: The Rail Gun• Conducting projectile of length 2cm,

mass 10g carries constant current 100A between two rails.

• Magnetic field B = 100T points outward.

• Assuming the projectile starts from rest at t = 0, what is its speed after a time t = 1s?

B I L

• Force on projectile: F= ILB (from F = iL x B)• Acceleration: a = iLB/m (from F = ma)• v(t) = iLBt/m (from v = v0 + at) = (100A)(0.02m)(100T)(1s)/(0.01kg) = 2000m/s 

= 4,473mph = MACH 8!

projectile

rails

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

Eprrr

×=τ

-Q

θQE

QE

+Q

p=Qa

Brrr

×=μτ

nNiA ˆ)(=μr