Transcript Physics 261 - Purdue University

Lecture 14-1
Magnetic Monopoles
• Does there exist magnetic charge, just like electric charge? An entity which
carried such magnetic charge would be called a magnetic monopole (having
+ or - magnetic charge).
•
How can you isolate this magnetic charge? Try cutting a bar magnet in half.
• In fact no attempt has been successful in finding magnetic monopoles in nature.
Lecture 14-2
Magnetic Field B
• Magnetic force acting on a moving charge q depends on q, v.
Vary q and v in the presence of a given magnetic field
and measure magnetic force F on the charge. Find:
F v
F varies sinusoidally as
F  qv
direction of v is changed
F  qv  B
This defines B.
(q>0)
direction by Right Hand
Rule. B is a vector field
F  v, B F  qvB sin 
F
N
N



 T (tesla )
 B 
qv C  m / s A  m
1 T = 104 gauss (earth magnetic
field at surface is about 0.5 gauss)
vB
If q<0
Lecture 14-3
Magnetic Force on a Current
A
• Consider a current-carrying wire in the
presence of a magnetic field B.
• There will be a force on each of the charges
moving in the wire. What will be the total force
dF on a length dl of the wire?
• Suppose current is made up of n charges/volume
each carrying charge q < 0 and moving with
velocity v through a wire of cross-section A.
• Force on each charge =
• Total force =
• Current =
qv  B
dF  n A(dl ) qv  B
I  n Av q
For a straight length of wire L carrying a current I,
the force on it is:
dF  Idl  B
F  IL  B
Lecture 14-4
Cyclotron
• "Magnetic Resonance Accelerator"
• "Dees" in constant magnetic field B
• Alternating voltage V is applied between the
Dees at the orbital frequency f:
v2
qvB  m
r
mv
r
 const.
qB
qB
2 r 2 m
qB
  2 f 
m
• Particle will acquire an additional kinetic energy T
= qV each time it crosses the gap (ie twice per
revolution.. E=0 in Dees!).
f 
v

mv
increases as v does
r
Problems  synchrotron
qB
http://www.kgv.net/amorrison/assets/animations/cyclotron.swf
Lecture 14-5
DOCCAM 2
FORCE BETWEEN TWO WIRES
Lecture 14-6
Synchrotron
mv
r
qB
r is the same since B increases as v does
Lecture 14-7
More complicated situations?
v is not perpendicular to B
Also non-uniform B
magnetic
bottle
helical motion (spiral)
electron in magnetic field
Van Allen belts
Lecture 14-8
Polar Light
High energy particles leaked out of the belt and interact with the
earth atmosphere.
Lecture 14-9
Warm-up quiz 1
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?
a)
b)
c)
d)
e)
Increase E
Increase B
Turn B off
Turn E off
Nothing
http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/magnetic/applet.html
http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/exb_thomson/applet.html
Lecture 14-10
Magnetic Force on a Current Loop
Force on closed loop current in uniform B?
– Force on top path cancels force on
bottom path (F = IBL)
– Force on right path cancels force
on left path. (F = IBL)
F  I L B
loop


 I L  B
 loop 
0
closed loop
Uniform B is exerts no net force
on closed current loop.
Lecture 14-11
Magnetic Torque on a Current Loop
Definition of
torque:
  rF
abut a chosen point
• If B field is parallel to plane of
loop, the net torque on loop is 0.
B
n




• If B is not zero, there is
net torque.
magnetic moment direction
n
so that n is
twisted to
align with B
Lecture 14-12
DOCCAM 2
ELECTRIC MOTOR
Lecture 14-13
Calculation 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:
b
τ   r  F  2  sin   F
2
F  IaB
  Iab  B  sin 
 IAB sin 
area of loop
• Define magnetic dipole moment by
  IAn
  B
where n is normal to the
loop with RHR along I.
 
  p E

Lecture 14-14
Example 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:
d    (r )(dr)  ( r 2 )  n
dI
1
1
   R 2  R 2   QR 2 
4
4
1
L  I m   mR 2 
2
Q

L
2m

R

   d     r 3dr n
1
  R 4 
4
0
Lecture 14-15
Potential 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
B
x
F

F
.

θ
U   τdθ

90
U   μB sin θdθ
90
Therefore,
U  μB cos θ 90  U   μB cos θ 
θ
U    B
Lecture 14-16
Potential Energy of Dipole Illustrated

B

x
B
B

x
x
=0
 = B X
=0
U = -B
U=0
U = B
min. energy
max torque
negative work
max. energy
positive work
(by YOU)
Lecture 14-17
Hall 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
perpendicular to both applied E
and B.
 Determines the sign and
number of carriers.
EH
EH
 Measures B.
Lecture 14-18
Carrier Sign and Density from Hall Effect
Sign and density of charge carrier
is determined at equilibriumHall
voltage
 q E H   q vd  B  0
EH
EH EH d VH
vd 


B
Bd
Bd
and
J
I
n

vd q vd qA
EH
BId
BI
n 

VH qA VH q  A / d 
VH  B
for a given
current I and n
Lecture 14-19
PHYS 241 10:30 Quiz 2
An proton (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 E
b. Turn B off
c. Turn E off
d. Slow down the electron
e. None of the above
Lecture 14-20
PHYS 241 11:30 – Quiz 2
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 E
b. Increase B
c. Turn B off
d. Slow down the proton
e. None of the above
Lecture 14-21
PHYS241 - Quiz C
A proton (charge +e) comes horizontally into a region of
perpendicularly crossed, uniform E and B fields as shown.
In this region, it deflects downward as shown. What can
you do to change the path so it remains horizontal through
the region?
a. Increase E
b. Turn B off
c. Turn E off
d. Slow down the electron
e. Increase B
Lecture 14-22
READING QUIZ 1
The Hall effect is the action of a magnetic field B on the carriers of a current
I = EL/R in a solid rectangular sample with area A = length L x width w.
A magnetic field is perpendicular to the area A and the current I = EL/R flows
parallel to the length L. Which of the following statements is incorrect.
A| The Hall voltage depends on the drift velocity of the charged carriers.
B| The Hall voltage depends on the length L of the solid sample.
C| The Hall voltage depends on the magnitude of the magnetic field.
D| The Hall voltage depends on the width w of the sample.