Magnetism

Magnetic Force

Magnetic Force Outline

• Lorentz Force

• Charged particles in a crossed field

• Hall Effect

• Circulating charged particles

• Motors

• Bio-Savart Law

Class Objectives

• Define the Lorentz Force equation.

• Show it can be used to find the magnitude and direction of the force.

• Quickly review field lines.

• Define cross fields.

• Hall effect produced by a crossed field.

• Derive the equation for the Hall voltage.

Magnetic Force

• The magnetic field is defined from the Lorentz Force Law, BvqEqF

Magnetic Force

• The magnetic field is defined from the Lorentz Force Law,

• Specifically, for a particle with charge q moving through a field B with a velocity v,

• That is q times the cross product of v and B.

BvqEqF

BvqF

Magnetic Force

• The cross product may be rewritten so that,

• The angle is measured from the direction of the velocity to the magnetic field .

• NB: the smallest angle between the vectors!

sinvBqF

v B

v x BB

v

Magnetic Force

Magnetic Force

• The diagrams show the direction of the force acting on a positive charge.

• The force acting on a negative charge is in the opposite direction.

+

-

v

F

F

B

Bv

Magnetic Force

• The direction of the force F acting on a charged particle moving with velocity v through a magnetic field B is always perpendicular to v and B.

Magnetic Force

• The SI unit for B is the tesla (T) newton per coulomb-meter per second and follows from the before mentioned equation .

• 1 tesla = 1 N/(Cm/s)

Bvq

F

sin

Magnetic Field Lines

Review

Magnetic Field Lines

• Magnetic field lines are used to represent the magnetic field, similar to electric field lines to represent the electric field.

• The magnetic field for various magnets are shown on the next slide.

Magnetic Field Lines

Crossed Fields

Crossed Fields

• Both an electric field E and a magnetic field B can act on a charged particle. When they act perpendicular to each other they are said to be ‘crossed fields’.

Crossed Fields

• Examples of crossed fields are: cathode ray tube, velocity selector, mass spectrometer.

Crossed Fields

Hall Effect

Hall Effect

• An interesting property of a conductor in a crossed field is the Hall effect.

Hall Effect

• An interesting property of a conductor in a crossed field is the Hall effect.

• Consider a conductor of width d carrying a current i in a magnetic field B as shown.

i i

d

xx x x

x xxx

xxxx

xxxx

B

Dimensions:

Cross sectional area: ALength: x

Hall Effect

• Electrons drift with a drift velocity vd as shown.

• When the magnetic field is turned on the electrons are deflected upwards.

i i

d

xx x x

x xxx

xxxx

xxxx

B

-vd

FB

FB

Hall Effect

• As time goes on electrons build up making on side –ve and the other +ve.

i i

d

xx x x

x xxx

xxxx

xxxx

B

-vd

- - - - -

+ + + + +High

Low

FB

Hall Effect

• As time goes on electrons build up making on side –ve and the other +ve.

• This creates an electric field from +ve to –ve.

i i

xx x x

x xxx

xxxx

xxxx

B

-vd

- - - - -

+ + + + +High

Low

FB

EFE

Hall Effect

• The electric field pushed the electrons downwards.

• The continues until equilibrium where the electric force just cancels the magnetic force.

i i

xx x x

x xxx

xxxx

xxxx

B

-vd

- - - - -

+ + + + +High

Low

FB

EFE

Hall Effect

• At this point the electrons move along the conductor with no further collection at the top of the conductor and increase in E.

i i

xx x x

x xxx

xxxx

xxxx

B

-vd

- - - - -

+ + + + +High

Low

FB

EFE

Hall Effect

• The hall potential V is given by, V=Ed

Hall Effect

• When in balance,EB FF

BeveE d

neA

ivwhere d

Hall Effect

• When in balance,

• Recall,

EB FF BeveE d

neA

ivwhere d

dt

dqi

dxneAdq

dx

A

A wire dt

dxneAi dvneA

Hall Effect

• Substituting for E, vd into we get,BeveE d

Vle

Bin

d

Alwhere

A circulating charged particle

Magnetic Force

• A charged particle moving in a plane perpendicular to a magnetic field will move in a circular orbit.

• The magnetic force acts as a centripetal force.

• Its direction is given by the right hand rule.

Magnetic Force

Magnetic Force

• Recall: for a charged particle moving in a circle of radius R,

• As so we can show that,

R

mvFB

2

R

mvqvB

2

qB

mvR

qB

mT

2

m

qBf

2,

m

qB,

Magnetic Force on a current carrying wire

Magnetic Force

• Consider a wire of length L, in a magnetic field, through which a current I passes.

x x

x

x

xx x

xI

B

Magnetic Force

• Consider a wire of length L, in a magnetic field, through which a current I passes.

• The force acting on an element of the wire dl is given by,

x x

x

x

xx x

xI

B

BLIdFd B

Magnetic Force

• Thus we can write the force acting on the wire, BIdLdFB

L

B dLBIF0

BILFB

Magnetic Force

• Thus we can write the force acting on the wire,

• In general,

BIdLdFB

L

B dLBIF0

BILFB

sinBILFB

Magnetic Force

• The force on a wire can be extended to that on a current loop.

Magnetic Force

• The force on a wire can be extended to that on a current loop.

• An example of which is a motor.

Interlude

Next….

The Biot-Savart Law

Biot-Savart Law

Objective

• Investigate the magnetic field due to a current carrying conductor.

• Define the Biot-Savart Law

• Use the law of Biot-Savart to find the magnetic field due to a wire.

Biot-Savart Law

• So far we have only considered a wire in an external field B. Using Biot-Savart law we find the field at a point due to the wire.

Biot-Savart Law

• We will illustrate the Biot-Savart Law.

Biot-Savart Law

• Biot-Savart law:

rldr

IBd ˆ

4 20

20

4

sin

r

IdldB

Biot-Savart Law

• Where is the permeability of free space.

• And is the vector from dl to the point P.

0ATm /104 7

0

r̂

Biot-Savart Law

• Example: Find B at a point P from a long straight wire.

l

Biot-Savart Law

• Sol: rldr

IBd ˆ

4 20

20

4

sin

r

IdldB

l

Biot-Savart Law

• We rewrite the equation in terms of the angle the line extrapolated from makes with x-axis at the point P.

• Why?

• Because it’s more useful. l

r̂

Biot-Savart Law

• Sol:

• From the diagram,

• And hence

rldr

IBd ˆ

4 20

20

4

sin

r

IdldB

180

90 l

Biot-Savart Law

• Sol:

• From the diagram,

• And hence

rldr

IBd ˆ

4 20

20

4

sin

r

IdldB

180

90 90sinsin cos

l

ABBABA cossincossinsin

Biot-Savart Law

• Hence,

• As well,

• Therefore,

20

4

cos

r

IdldB

x

ltan

r

xcos 22 lxr

x

dIdB

4

cos0

l

Biot-Savart Law

• For the case where B is due to a length AB,

A

B

d

x

idBB

B

cos4

0

0

sinsin

40 x

i

Biot-Savart Law

• For the case where B is due to a length AB,

• If AB is taken to infinity,

A

B

d

x

idBB

B

cos4

0

0

sinsin

40 x

i

x

iB

2

0