Magnetic Forces and Fields

Magnetic Force

Right Hand Rule: Cross Product

BF qv B

Magnetic Force on a Wire

B

B

F Il B

dF Idl B

FB and Circular Motion

2

2

2 2

vma m

r

vqvB m

rr m

Tv qB

Velocity SelectorOnly charges with the proper velocity will pass through undeflected.

so and E B

EF F qE qvB v

B

Mass Spectrometer

Particles are ionized and sent through velocity selector. Radius depends on mass if charge and velocity are equal.

mvrqB

The Cyclotron

Torque on a LoopDefine a unit vector perpendicular to the plane of the loop.

Torque on a Loop

2 so sin

so

bB B

B B

r F F

F Il B F IaB

For each side of length a:

No force is produced on sides of length b since current is parallel to B. Net force on loop is zero but torque will be produced due to forces on the other sides. Calculating torques about the midpoint of the loop,

Since there are two sides:

sin

the area of the loop

For a loop of N turns:

sin

IabB

ab A

NIAB

The Hall EffectCurrent through conducting strip causes charge separation and establishes potential difference due to magnetic force on moving charges.

If mobile charges are positive, top of strip is positive due to upward force.

The Hall EffectIf mobile charges are negative, force is still upward by top of strip will be negative, allowing for identification of mobile charge type.

If potential difference is measured, magnetic field strength or drift velocity can be calculated.

Field of a Bar Magnet

Biot-Savart Law

• Similar to Coulomb’s Law for electrostatics• Define differential current element Ids• Integrate over current distribution to find

total magnetic field

Ampere’s Law

• With the proper symmetry, Ampere’s Law can be used in place of Biot-Savart

• Similar to Gauss’ Law for electrostatics• B must be constant along the path of

integration and parallel to the differential of length, dl