Download lecture12

5. Magnetic forces on current

vd
A
N – number of charge carriers in the volume lA
n – density of charge carriers (in metals it is density of electrons)
l
N
n
lA

vd – drift velocity

 
F  qv  B

 
Ftot  Nqvd  B
 
 nlAqvd  B
 
 nqAvd l  B
  
F  Il  B
 

F  Il  B
F  IlB sin 
F  IlB sin 
N  nlA
 
lvd  l vd
nqAvd  I


 - angle between vd and B
Example: A straight wire carrying a current is placed in a region containing a
magnetic field. The current flows in the +x direction. There are no magnetic
forces acting on the wire. What is the direction of the magnetic field?
F 0 
sin   0    0, or   180
1
Example: A horizontal wire carries a current and is in a vertical magnetic field.
What is the direction of the force on the wire?
1) left
2) right
I
3) zero
4) into the page
B
5) out of the page
Example: A vertical wire carries a current and is in a vertical magnetic field.
What is the direction of the force on the wire?
1) left
I
2) right
3) zero
4) into the page
B
5) out of the page
2
6. Current loop in magnetic field
Example 1: A rectangular current loop is in a uniform magnetic field.
What is the direction of the net force on the loop?
1)
2)
3)
4)
5)
+x
+y
zero
-x
-y
Using the right-hand rule, we find that each
of the four wire segments will experience a
force outwards from the center of the loop.
Thus, the forces of the opposing segments
cancel, so the net force is zero.
z
x
B
y
Example 2: A wire loop is in a uniform magnetic field.
Current flows in the wire loop, as shown.
What does the loop do?
1)
2)
3)
4)
5)
moves to the right
moves up
remains motionless
rotates
moves out of the page
There is no magnetic force on the top and bottom legs, since they are parallel to the B
field. However, the magnetic force on the right side is into the page, and the magnetic
force on the left side is out of the page. Therefore, the entire loop will tend to rotate.
This is how a motor works !!
3
Example: If there is a current in the loop in the direction shown, the loop will:
1) move up
F
2) move down
3) rotate clockwise
S
N
4) rotate counterclockwise
N
B
S
5) both rotate and move
B field out of North
B field into South
F
At the North Pole, the magnetic field points to the right and the current points out
of the page. The right-hand rule says that the force must point up. At the south
pole, the same logic leads to a downward force. Thus the loop rotates clockwise.
4
Wire and Loop with Electric Current in Magnetic Field
Battery
B
I
Axis of rotation
Fm
N
B
S
I
Fm
Fm
I
Magnetic field
Loop
The loop will rotate (clockwise)
S
N
Fm
N
I
S
N
S
B
5
6a. Current loop and magnetic moment
B
Fa  IaB
Fb '
Fa '
I
a
Fb  IbB sin  '



Fa   Fa '


F  0
Fb   Fb '
n̂
I
ab  A
b
Fa
Fb
b
  2 Fa sin   IaBb sin   IAB sin 
2
n̂
b

'

Fa 

B
N – number of loops in the coil
Definition of magnetic

dipole moment: M  NIAnˆ
  MB sin 
 N  N 1

 - angle between th e perpendicu lar to the coil, nˆ , and B


 angle between M and B

 angle between th e direction of the current in b and Fa
6
  B sin 
 2

 2


Example 1 : A current-caring circular coil is in uniform magnetic field. The maximum
magnitude of the torque on the coil occurs when the angle between the field and
magnetic moment is
0
45°
90°
135°
180°
Example 2: A coil is oriented initially with its magnetic dipole moment antiparallel to
a uniform magnetic field. In this orientation torque  is equal to:
A)  = 0
B)  = max
This is unstable equilibrium
C) 0<  < max
7
Example:
N  10
A  r 2
r  0.10m
  30 
I  3.00 A
B  2.00T
M ?
 ?
M  NIA
M  10  3.00 A  0.10m2  0.94 Am2
  MB sin 
  0.94 A  m2  2.00T sin 30  0.94 N  m
8