Lecture 14-Lecture 14-11Magnetic Field B

• Magnetic force acting on a moving charge q depends on q, v.

F qv B ������������������������������������������

(q>0)

v B������������� �

If q<0

A

F IL B

Lecture 14-Lecture 14-22Hall Effect

• A conducting strip in crossed E and B fields

• Applied E along the strip leads to a charge buildup on the sides of the strip and thus an electric field EH develops to both applied E and B.

Determines the sign and number of carriers.

Measures B.

EH

EH

Lecture 14-Lecture 14-33Carrier Sign and Density from Hall Effect

Sign and density of charge carrier is determined at equilibrium

0H d

H H Hd

q E q v B

E E d Vv

B Bd Bd

������������������������������������������

and

/

d d

H H

J In

v q v qA

BId BIn

V qA V q A d

Hall voltage

for a given current I and nHV B

EH

EH

Lecture 14-Lecture 14-44Cyclotron

• "Dees" in constant magnetic field B

• Alternating voltage V is applied between the Dees at the orbital frequency f:

• "Magnetic Resonance Accelerator"

• Particle will acquire an additional kinetic energy T = qV each time it crosses the gap (ie twice per revolution.. E=0 in Dees!).

mvr

qB increases as v does

problems synchrotron

2 2

2

v qBf

r mqB

fm

2

.

vqvB m

rmv

r constqB

Lecture 14-Lecture 14-55Synchrotron

mvr

qB R is the same since B increases as v does

Lecture 14-Lecture 14-66More complicated situations?

helical motion (spiral)

v is not perpendicular to B

Van Allen belts

Also non-uniform B

magnetic bottle

Lecture 14-Lecture 14-77Polar Light

High energy particles leaked out of the belt and interact with the earth atmosphere.

Lecture 14-Lecture 14-88 Warm-up

a) Increase Eb) Increase Bc) Turn B offd) Turn E offe) Nothing

An electron (charge -e) comes horizontally into a region of perpendicularly crossed, uniform E and B fields as shown. In this region, it deflects upward as shown. What can you do to change the path so it remains horizontal through the region?

http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/exb_thomson/applet.html

http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/magnetic/applet.html

Lecture 14-Lecture 14-99Magnetic Force on a Current Loop

– Force on top path cancels force on bottom path (F = IBL)

– Force on right path cancels force on left path. (F = IBL)

loop

loop

I L B

I L B

F

�������������� ����������������������������

����������������������������

Force on closed loop current in uniform B?

Uniform B exerts no net force on closed current loop.

closed loop

0

Lecture 14-Lecture 14-1010Magnetic Torque on a Current Loop

• If B field is to plane of loop, the net torque on loop is also 0.

r F ������������� �Definition of

torque:

abut a chosen point

• If B is not , there is net torque.

B

Lecture 14-Lecture 14-1111Calculation of Torque

• Suppose the coil has width b (the side we see) and length a (into the screen). The torque about the center is given by:

2 sin2

bτ r F F

F IaB

sin

sin

Iab B

IAB

area of loop

• Define magnetic dipole moment by

IAn ��������������

B �������������������������� ��

where n is normal to the loop with RHR along I. Ep

Lecture 14-Lecture 14-1212Example of Magnetic Moment Calculation

A thin non-conducting disk of mass m and uniform surface charge density rotates with angular velocity as shown. What is the magnetic moment?

n

mag. moment of the ring shown:

2( )( ) ( )d r dr r n ��������������

dI

3

0

41

4

Rd r dr n

R

����������������������������

�������������� 2 2 21 1

4 4RR QR

������������������������������������������

21

2mL I mR ������������������������������������������

2

QL

m ����������������������������

Lecture 14-Lecture 14-1313Potential Energy of Dipole

• Work must be done to change the orientation of a dipole (current loop) in the presence of a magnetic field.

• Define a potential energy U (with zero at position of max torque) corresponding to this work.

θ

τdθU90

θ

θdθμBU90

sin

Therefore,

θθμBU 90cos θμBU cos

Bx

F

F

.

BU

Lecture 14-Lecture 14-1414Potential Energy of Dipole Illustrated

B

x

Bx

B

x

= 0

U = -B

= 0

U = B

negative work

=B

X

U = 0

positive work (by YOU)

min. energy max. energymax torque

Lecture 14-Lecture 14-1515PHYS241 - Quiz A

An electron (charge e) comes horizontally into a region of perpendicularly crossed, uniform E and B fields as shown. In this region, it is deflected upward as shown. What can you do to change the path so it deflects downward instead through the region?

a. Increase Eb. Turn B offc. Decrease E d. Slow down the electrone. None of the above

Lecture 14-Lecture 14-1616PHYS241 - Quiz B

A proton (charge +e) comes horizontally into a region of perpendicularly crossed, uniform E and B fields as shown. In this region, it goes straight without deflection. What can you do to change the path so it deflects upward through the region?

a. Increase Eb. Increase Bc. Turn B offd. Slow down the protone. None of the above

Lecture 14-Lecture 14-1717PHYS241 - Quiz C

A proton (charge +e) comes horizontally into a region ofperpendicularly crossed, uniform E and B fields as shown.In this region, it deflects downward as shown. What canyou do to change the path so it remains horizontal throughthe region?

a. Increase Eb. Turn B offc. Turn E offd. Slow down the electrone. Increase B