Physics 2102Physics 2102Lecture 15Lecture 15Magnetic fieldsMagnetic fields

Physics 2102

Jonathan Dowling

Star Quake on a

Magnetar!“I’ll be back….

Aurora Borealis

Use of MagneticFields in YourEveryday Life!

Magnetic FieldsMagnetic FieldsElectric fields are created:• microscopically, by electric charges (fields) of elementary particles (electrons, protons)• macroscopically,by adding the field of many elementary charges of the same sign

Magnetic fields are created :• microscopically, by magnetic “moments” of elementary particles (electrons, protons, neutrons)• macroscopically, by

• adding many microscopic magnetic moments(magnetic materials); or by• electric charges that move (electric currents)

We know that an electric fields exists because it accelerates electric charges, with a force independent of the velocityof the charge, proportional to the electric charge: FE = qE

We know that a magnetic field exists because it accelerateselectric charges in a direction perpendicular to the velocityof the charge, with a magnitude proportional to the velocityof the charge and to the magnitude of the charge: FB= q v x B

Magnetic forces are perpendicular to both the velocity of chargesand to the magnetic field (electric forces are parallel to the field).

Since magnetic forces are perpendicular to the velocity,they do no work! (W=F · r)

Speed of particles moving in a magnetic field remains constantin magnitude, the direction changes. Kinetic energy is constant!(no work).

Magnetic vs. ElectricMagnetic vs. ElectricForcesForces

Circular Motion: Since magnetic force is transverse to motion,the natural movement of charges is circular.

B into blackboard.

vF motioncircular for

2

r

vmmaF ==

r

mvBvq

2

Therefore =qB

mvr =

In general, path isa helix (component ofv parallel to field isunchanged).

r

ExampleExampleTwo charged ions A and B travelingwith a constant velocity v enter a boxin which there is a uniform magneticfield directed out of the page. Thesubsequent paths are as shown. Whatcan you conclude?

qB

mvr =

Same speed and B for both masses. So: ion with larger mass:charge ratio (m/q) moves in circle of larger

radius. But that’s all we know!

(a) Both ions are negatively charged.

(b) Ion A has a larger mass than B.

(c) Ion A has a larger charge than B.

(d) None of the above.

v

v

A

B

Cathode Ray Tube (Old TVs & Computer Monitors)

Hot cathode emits electrons

Get accelerated by positive plateMight be deflected using plates

Produce point of light on screen.

In a magnetic field: B

vBvrr

!

Fe

Dot shifts sideways.

Aurora borealis(northern lights)

Synchrotron

Linear accelerator (long).

Fermilab,Batavia, IL (1km)

Suppose you wish to accelerate chargedparticles as fast as you can.

Examples of Motion in Magnetic FieldsExamples of Motion in Magnetic Fields

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

dv

Litiq ==

LBvqF

d

rrr!=

BLiBq

LiqF

rrrr

r!=!=

BLiFrrr

!=

BLdiFdrrr

!=

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

ExampleExample

iLBFF ==31

!iBRdiBdLdF ==

By symmetry, F2 will only have a vertical component,

iBRdiBRdFF 2)sin()sin(00

2 !! ===""

###

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

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

L LR Rand 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 current100A between two rails.

• Magnetic field B = 100T pointsoutward.

• Assuming the projectile starts fromrest at t = 0, what is its speed after atime 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

Rail guns in the “Eraser” movie"Rail guns are hyper-velocity weapons that shoot aluminum or clay rounds atjust below the speed of light. In our film, we've taken existing stealth technologyone step further and given them an X-ray scope sighting system," notes directorRussell. "These guns represent a whole new technology in weaponry that is stillin its infancy, though a large-scale version exists in limited numbers on

battleships and tanks. They haveincredible range. They can piercethree-foot thick cement walls andthen knock a canary off a tin canwith absolute accuracy. In our film,one contractor has finally developedan assault-sized rail gun. Weresearched this quite a bit, and thetechnology is really just around thecorner, which is one of the excitingparts of the story."

Warner Bros., production notes, 1996.http://movies.warnerbros.com/eraser/cmp/prodnotes.html#tech

Also: INSULTINGLY STUPID MOVIE PHYSICS: http://www.intuitor.com/moviephysics/

Torque on a Current Loop: Principle behind electric motors.

Net force on current loop = 0

iaBFF ==31

)sin(1

!FF ="

)sin(!" 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,µ

r

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

B

rrr!= µ"

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

Electric vs. Magnetic DipolesElectric vs. Magnetic Dipoles

Eprrr

!="

-Q

θQE

QE

+Q

p=Qa

B

rrr!= µ"

nNiA ˆ)(=µr