Transcript Lec15

Lecture 15-1
What we learned from last class
Hall effect:
Determines the sign and number
of carriers.
 Measures B.
• Define magnetic dipole moment by
  IAn
  B
where n is normal to the
loop with RHR along I.
U    B
Lecture 15-2
Van Allen belts
Lecture 15-3
Sources of Magnetic Fields
• Moving point charge:
μ0 q v  r
dB 
4π r 2
 N
0  4  10  2 
A 
7
Permeability constant
• Bits of current:
I
μ0 I d l  r
dB 
4π r 2
Biot-Savart Law
The magnetic field “circulates”
around the wire.
Lecture 15-4
Magnetic field due to a current loop
Principle of superposition:
B
0
4

Id l  r
r2
At the center, dl  r  dl  sin  x  dl x (   / 2)
I
I
B  0 2 x  dl  0 x
4 R
2R
On axis generally,
0 I dl 0 I dl

2
4 r
4 R 2  x 2
I
R
 dBx  dB  sin   0
dl
3/ 2
2
2
4  R  x 
dB 
0 I
R3
 Bx  dB
dBxx 
 2
2 R ( R  x 2 )3/ 2
Lecture 15-5
Circular Loop Current as a Magnetic Dipole
Lecture 15-6
Magnetic Field of Circular Arc Current
0 ids sin 90 0 ids
dB 

2
4
R
4 R 2
0i
ds


0i 
B 


2 R 2 R 2 R 2
I runs clockwise in the
closed loop wires below:
What is B at center?
Lecture 15-7
PHYS241 – warm-up
A circular current loop lies in the plane perpendicular to
this sheet with its axis along the x-direction, and produces
magnetic field B as shown. What is the direction of the
current at the top end of this loop?
a. Out of the sheet
b. Into the sheet
c. Along +x axis
d. Along x axis
e. Current is zero.
x
Lecture 15-8
Gauss’s Law for Magnetism
sources
Gauss’s Law
E   E  d a 
s
qs
0
Gauss’s Law for Magnetism
No sources
B   B  d a  0
s
Lecture 15-9
Magnetic field of a solenoid
L
• A constant magnetic field could be
produced by an infinite sheet of current.
In practice, however, it is easier and
more convenient to use a solenoid.
• A solenoid is defined by a current I
flowing through a wire that is wrapped n
turns per unit length on a cylinder of
radius R and length L.
R
Lecture 15-10
Magnetic field of a solenoid (continued)
Contribution to B at origin from length dx
0 I
2 R 2
dBx 
 2
 n  dx
2 3/ 2
4 ( R  x )
one turn
# turns in
length dx
x
0
dx
2
 Bx 
 2 R nI  
x ( R 2  x 2 ) 3/ 2
4
2
1

1
x2

 0nI

2
2

2
 x2  R


2
2 
x1  R 
x1
As x1  , x2   (or for L>>R)
Bx  0nI
(half at ends)
Lecture 15-11
Solenoid’s B field synopsis
• Long solenoid (R<<L):
B inside solenoid
// to axis
B outside solenoid
nearly zero
(not very close to the ends or wires)
Lecture 15-12
RHIC STAR Experiment
STAR
Lecture 15-13
Magnetic field of a Straight Current
μ0 I dx
μ0 I dx
dB 
sin  
cos 
2
2
4π r
4π r
y
2
r
r2
 
2
d
x  R tan θ , dx  R sec  dθ  R   d 
R
R
2
μ0 I cos 
 B   dB  
d
4π R
1 I
0 I
B 
 sin 2  sin 1 
4 R
1   / 2,2   / 2
Infinite straight current
0 2 I 0 I
B

4 R 2 R
http://falstad.com/vector3dm/
Lecture 15-14
Two Perpendicular Currents
B
FB2,1
FB 1,2
I2
B
I1
I3
FB 1,3
Lecture 15-15
PHYS241 - Quiz A
A circular current loop lies on the xy-plane as shown,
where the current is clockwise as seen from the positive z-axis.
What is the direction of the B field at point A?
z
a. Along +x axis
b. Along +y axis
c. Along +z axis
d. Along z axis
e. Along  x axis
y
I
x
A
Lecture 15-16
PHYS241 - Quiz B
A circular current loop lies on the xy-plane as shown,
where the current is clockwise as seen from the positive zaxis. What is the direction of the B field at point A?
z
a. Along +x axis
b. Along +y axis
c. Along +z axis
d. Along z axis
e. Along  x axis
y
I
x
A
Lecture 15-17
PHYS241 - Quiz C
A circular current loop lies on the xy-plane as shown,
where the current is clockwise as seen from the positive zaxis. What is the direction of the B field at point A?
z
a. Along +x axis
b. Along +y axis
c. Along +z axis
d. Along z axis
e. Along  x axis
y
A
I
x