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 Lorentz Force Law,

F

 

q E

 

q v

 

Magnetic Force Lorentz Force Law,

F

 

q E

 

q v

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

F

 

q v

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

Magnetic Force • The cross product may be rewritten so that,

F



q vB

sin     of the velocity to the magnetic field .

• NB: the smallest angle between the vectors!

v x B B  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.

B F v B + F v

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)

q v 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.

x x x x Dimensions: d x x x x i x x x x i Cross sectional area: A Length: x x

B

x x x

Hall Effect • Electrons drift with a drift velocity v d shown.

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

x x x i d x x x v d x x F B x x x x i x

B

x x x F B

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

i d x x x x v d x x F B x x x

B

x x x x x x Low i High

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

E

x x x v d x x F B F E x x x x x x Low i High x

B

x x x

Hall Effect • The electric field pushed the electrons downwards.

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

E

x x x x Low i x v d x x F B F E x x x x i High x

B

x x x

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

E

x x x x v d x x F B F E x x x

B

x x x x x x Low i High

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

Hall Effect • When in balance, 

eE



ev d B where v d



i neA F B



F E

Hall Effect • When in balance, 

eE



ev d B where v d



i neA F B



F E

• Recall,

dq i

 

i

    

dx dx dt dq dt

  

v d

A dx A wire

Hall Effect • Substituting for E, v d

n



Bi Vle where l



A d ev d

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 2 circle of radius R,

F B



mv R



qvB



mv R

2  • As so we can show that, ,

f R

 

mv qB qB

2 

m

,  

qB m T

 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 B x x I x x x x

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

x x B x x I x x x x • The force acting on an element of the wire dl is given by, 

d F B

 

Id L



B

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

dF B



BIdL



F B



BI

0 

L dL



F B



BIL

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

dF B



BIdL



F B



BI

0 

L dL



F B



BIL

• In general,

F B



BIL

sin 

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: 

d B

  4  0

r I

2 

d l

 

r

ˆ 

dB

  0

Idl

4 

r

sin 2 

Biot-Savart Law • Where is the permeability of free space.

 0  4    0 10  7

Tm

/

A r

ˆ

dl

to the point P.

Biot-Savart Law • Example: Find B at a point P from a long straight wire.

l

Biot-Savart Law • Sol: 

d B

  4  0

r I

2 

dB

  0

Idl

4 

r

sin 2  l

Biot-Savart Law • We rewrite the equation in terms of the

r

ˆ with x-axis at the point P.

• Why?

• Because it’s more useful.

l

Biot-Savart Law • Sol: 

d B

  4  0

r I

2   • From the diagram,     180   

dB

  0

Idl

4 

r

sin 2  • And hence     90  l   

Biot-Savart Law • Sol: 

d B

  4  0

r I

2   • From the diagram,     180   

dB

  0

Idl

4 

r

sin 2  • And hence     90   sin   sin    cos   90    sin 

A



B

  sin

A

cos

B

 sin

B

cos

A

 l    

Biot-Savart Law • Hence, • As well,

dB

  0

Idl

4 

r

cos  2 tan  

l x

cos  

r x r



x

2 

l

2 • Therefore,

dB

  0

I

cos  4 

x d

 l 

Biot-Savart Law • For the case where

B

B

  0 

B dB

 4  0

x i

  4 0 

x i

 sin     cos   sin  

d

 is due to a length AB,

A B

 

Biot-Savart Law • For the case where

B

B



B

 0

dB

  4 0 

x i

   cos 

d

 is due to a length AB,

A B

  4  0

x i

 sin   sin     • If AB is taken to infinity,

B

  2 0 

x i