Download PHYS_3342_110811

The graded exams are being returned today. You will have until the next class on
Thursday, Nov 10 to rework the problems you got wrong and receive 50% added credit.
Make sure you are in class as you will not have another opportunity to turn in the
reworked exam. I will be going over the answers in class on Thursday. This will also be
your only opportunity to ask for corrections/clarifications on any grading mistakes.
Magnetic Dipole in a Non-uniform Magnetic Field



F  (   ) B
y
B B
F   y
;
0
y y
F  0 (along y  axis)
B
0
y
F  0 (opposite to y  axis)
Magnetic Dipoles and How Magnets Work
The Direct-Current Motor
Magnetic Field of a Moving Charge
0 qv sin 
B
4 r 2

B
 
0 q v  r
4 r 2
Magnetic field of a point charge
moving with constant velocity

1

1
0
T

m
/
A
;



4

c
0

7
00 2
Example: Force between two moving protons
Find the ratio of electric and magnetic forces on the protons
1 q2
F
E
4
0 r2
0 qv
B
k
2
4
r

Magnetic field of the lower proton at
the position of the top one



F

qv
( )B
B

4
r
22

q
0 v
F
j
B
2
2
F
2 v
B


v 2
00
F
c
E
Magnetic Field of Current Element
The Biot-Savart law.
d
Q

n
q
A
d
l
f
l
o
w
w
i
t
h
v
e
l
o
c
i
t
y
v
d
For element of a (fine) wire:
 I dl  rˆ
dB  0
4 r 2
constant permeability of free space:
For the whole "circuit":
0 I dl  rˆ
B
4  r 2
For arbitrary distribution of charge flow:
 j(1)  rˆ12
B(2)  0 
dV1
2
4
r12
(rˆ12 is from point 1 to point 2)
Magnetic field around a straight wire
For the
fieldmagnitude
:
0I  sin dx
B

2

4  r
a
ad
[r 
; x  acot; dx 2 ] 
sin
sin 
0I 
0I

sin

d


4a 0
2a
(where
ais thedistance
fromthewire)
Magnetic Field of Two Wires
Field at points on the x-axis
to the right of point (3)

I
0
B

;
1
2

(
xd
)

I
0
B

;
2
2

(
xd
)

I
d
0
B

B

B

t
o
t
a
l
2 1
2 2

(
x

d)
Magnetic field outside of a conductor pair falls off more rapidly
Magnetic field of a circular arc
For the field magnitude at O :
0 I
B
4R 2
0 I
 ds  4R 2 R
0 I

4R

Magnetic Field of a Circular Current Loop
For field on the axis :
 I cos  ds
B ( x )  Bx ( x )  0  2
4 x  R 2
0 I
R
2R

2 3/ 2
2
4 ( x  R )

0 IR 2

2( x 2  R 2 )3 / 2 2 ( x 2  R 2 )3 / 2
[ x  R]

Falls off just as the electric field of
the electric dipole
Magnetic Field on the Axis of a Coil
Bx 
Bx 
0 NIR 2
2( x 2  R 2 )3/ 2
0 
2 ( x  R )
2
2 3/ 2
;
0 
  NIA
0 
2 x 3
The magnetic field of a (small) loop behaves “on the outside” like the electric field
of the electric dipole of the same orientation – that’s why “magnetic dipole”.
Magnetic force between two parallel conductors with currents
Magnetic field from conductor 2:
0 I
B2 
2 r
Magnetic force on conductor 1:
'

II
F1  I ' LB2  0 L
2 r
Absolutely the same magnitude is
for the magnetic force on conductor 2
but F1   F2
FB 0 II '

L
2 r
Currents in the same direction attract
Currents in opposite directions repel
Definition of 1 Ampere :
Identical current in two wires
separated by 1 m
is 1 Ampere
when the force per 1 meter
is 2 10 7 N/m
Example: Two straight, parallel, superconducting wires 4.5 mm apart carry
15,000 A current each in opposite directions
Should we carry about the mechanical strength of the wires?
F 0 II '

 104 N / m
L 2 r
Ampere’s Law

Circulation of B around a closed loop is 0 times
the total current through the surface bounded by the loop


 B d l 
 B dl  B  dl 
0 I
(2 r )  0 I
2 r


 B d l 
b
d
a
c
 B dl  B1  dl  ( B2 ) dl 
0 I
I
(r1 )  0 (r2 )  0
2 r1
2 r2
General Statement


 B d l  0 Iencl
 (Ampere's Law)
Magnetic fields add as vectors, currents – as scalars
Just as with the integral form of Gauss’s law, the integral form of
Ampere’s law is powerful to use in symmetric situations
Magnetic field around and inside a straight w ire
0 I 0
For path 1 : B  (2r )  0 I 0  B 
2r
0 I 0 r
r2
For path 2 : B  (2r )  0 I 0 2  B 
R
2R 2
Magnetic Field of a Solenoid
Wire wound around a long cylinder
produces uniform longitudinal field in
the interior and almost no field outside
For the path in an ideal solenoid:
BL  0nIL  B  0nI
(n turns of the coil per unit length)