magnetic interaction
There is interaction between a particle and other bodies which depends on
the charge of the particle, its position and its velocity (and its spin). We
call this interaction a magnetic interaction. Moving charged particles in
the body cause the magnetic interaction.
B

++
v
F
O
The region of space in
which magnetic forces
can be exerted is called
a magnetic field.
magnetic field vector


The magnetic field vector B at position r is a vector such that,
at this position, the magnetic force exerted on a particle with

charge q, moving with velocity v, would be

 
FB  qv  B
B
F
++
v
the cyclotron frequency



 FB q 
 qB  
a
  v
 v  B    
m m
 m
Recall that in a uniform circular
motion:
  
a =  v
B

++
v
F
O
In a uniform magnetic field, the
particle moves with constant
angular velocity along the
magnetic field


qB

m
work due to magnetic interaction
ds
+
v
F
 


dWB  FB  ds  FB  vdt  0
B
The magnetic work performed on the particle is zero.
(A magnetic field cannot change the speed of a particle.)
magnetic force "on a current"
B
Ids
dF
I

The (differential) force dF, exerted on a
(differential) segment, depends on the
current I in the wire, the size ds and
orientation sˆ of the segment, and the
magnetic field vector B at the location of
the segment:

 
dF  Id s  B


 
 
dF  qvdsˆ  B  nAds   nv dq  A   ds  B  Id s  B
magnetic moment

The magnetic moment of an object is a vector  such that


the magnetic torque
exerted on the object (about its
center of the mass) placed in the magnetic field B is
  
  B


Puzzle. What is the direction of
the magnetic moment
of a compass needle?
B
N
B
B
S
B
magnetic moment of a current loop
The magnetic moment of a wire loop carrying current depends
 
 loop.
a and the area
a A of the
current
 Iin the
 loop
on the
 1  2  3  4  Ib  nˆ  B  Ib  nˆ  B  0  0  IA  B
2 
2
  IA
nˆ

F1
b

B




a
F2
potential energy
The potential energy of an object in a magnetic field
depends on the magnetic moment of the object and the
magnetic field at its location
 
U    B


S
N
S
N
The Lorentz force
When the particle moves in the presence of both a magnetic
field and an electric field, the net force depends on both fields:


 
F  qE  qv  B
Example. The Hall effect
V
qvd B  qEH
VH  E H d  v d Bd
d
VH 
IB
nqA
+
+
+
+
+
+
+
FB
d
vd
+
_
FE
_
_
_
I
_
_
_
Maxwell's equations
. . . and God said:
Let . . .
  Q
 E  dA  
0
Gaussian
Surface
d B
 E  ds  
dt
Amperian
Loop
 
 B  dA  0
Gaussian
Surface
d E 

B

d
s



I




0 
0
dt


Amperian
Loop
. . . and there was light.
Gauss‘s law for magnetic fields
The net electric flux through any
The net magnetic flux through
closed (Gaussian) surface is
any closed (Gaussian) surface
proportional to the net charge
is equal to zero:
inside the surface:
 
  B dA q0in
E  dA 
Gausian
0
SSurface
N
Faraday's law of induction
N
B
E
The line integral of the electric
field vector around any closed path
equals the rate of change in the
magnetic flux through any surface
bounded by that path.
 
d B
E

d
s



dt
closed
loop
Ampere-Maxwell law
E
E
B
I
The circulation of the magnetic
field vector around any amperian
loop proportional the sum of the
total conduction current and the
displacement current through any
surface bounded by that path.
 
d E 


 B  ds   0  I   0
dt 

closed
loop
The proportionality coefficient is
called the permeability of free space.
displacement current
The rate of change in the electric field multiplied by the
permittivity of free space is called the displacement current
dE
Id   0
dt
I
Example:
Q
E
-Q
I
Id  0
d E
dt
Q
d 
 0  dQ

 0
I

dt
dt
Example:
Magnetic field of a long straight wire with current
I
v
ds
B
 0I
B
2 R
+
R
F
2 R
  2R
 0 I   B  d s   Bds  B  ds  2R  B
loop
0
0
the Biot-Savart law
dB
Ids
 0 Ids  r
dB 

4
r2
P
r
I
The (differential) magnetic field at
a certain position P produced by a
differential
element
carrying
electric current I depends on the
value of the current and the size
and orientation of the segment.
Example: infinite straight wire with current
I

0 Id s  rˆ
dB y 
 2 
4 r
z
y
dB
x
R
P
r
s

Ids


B  B y  0 IR  
4 -
R
s
2

3
2
 0 I sin   ds


2
4
r
0 I
1
R

 2 2
 ds 
2
2
4 R  s
R s

ds
2

 0 IR

4
R
ds
2
s
0
s

IR 
4
R 2 R 2  s2
2

3


2
0I

2 R
Interaction between two parallel current
B1
F21
I1
I2
a
1
l
The magnitude of the
magnetic force exerted on
segment l of a wire by the
other wire (infinite) is
2




0 l 
F21  I2 l  B1 
 I2  I1  aˆ  
2a

 
0 l 

 I2  aˆ I1  I1  I2  aˆ  
2a
0 l  
I1  I2  aˆ

2a
F21 
0l
 I1I2
2a
Parallel "currents" attract
and antiparallel repel.
magnetic field of a solenoid
The magnetic field outside
the solenoid is zero
Bout  0
The magnetic field inside
is uniform, its direction is
parallel to the axis, and the
magnitude depends on the
current and the number of
loops per length of the
solenoid
Bin  0nI
N
S
L
I
I
Magnetic properties of matter
When a substance is placed in a (external) magnetic field, its
molecules acquire a magnetic moment related to the external
field. This creates an additional magnetic field (internal).
 


B  B 0  Bm  1  B 0
B0
paramagnetics:  > 1
Bm
diamagnetics:
<1