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Biot-Savart Law
  0 i (ds  r )
dB 
3
4
r
 
B

d
S

0

The Field Produced by a Straight Wire
0 i
B
2 a
Field of a Current Carrying Loop
r
R
x
B
 0i
R2
2 (R2  x2 )
3
(along x)
2
0 M
x  R B 
2 x 3
Ampere’s Law
 
B

d
r


i
0

The field produced by an infinite wire
0 i
B
2 a
Problem 4
Consider a very long (essentially infinite), tightly wound
coil with n turns per unit length. This is called a solenoid.
Assume that the lines of B are parallel to the axis of the
solenoid and non-zero only inside the coil and very far
away. Also assume that B is constant inside. Find B inside
the solenoid if there is a current i flowing through it.
Problem 3
An infinitely long wire has 5 amps flowing in it. A
rectangular loop of wire, oriented as shown in the
plane of the paper, has 4 amps in it. What is the force
exerted on the loop by the long wire?
Induced EMF and Inductance
1830s Michael Faraday
Joseph Henry
Faraday’s Law of Induction
The induced EMF in a closed loop equals the negative of the
time rate of change of magnetic flux through the loop
d B
EMF  
dt
 
d B
 E  dr   dt
S
S
 
d B  B  dS  B dS  BdS cos 
There can be EMF produced in a
number of ways:
•
•
•
•
A time varying magnetic field
An area whose size is varying


A time varying angle between B and dS
Any combination of the above
R
From Faraday’s law: a time
varying flux through a
circuit will induce an EMF
in the circuit. If the circuit
consists only of a loop of
wire with one resistor, with
resistance R, a current
EMF
i
R
Which way?
Lenz’s Law: if a current is induced by some
change, the direction of the current is such
that it opposes the change.
 
d B
 E  dr   dt
The Moving Circuit
S
The magnetic field between the poles of the
electromagnet is uniform, but its magnitude is
increasing at the rate of 0.020 T/s. The area
of the conducting loop in the field is 120 cm2,
and the total circuit resistance, including the
meter and the resistor, is 5.0 Ώ. Find the
induced EMF and the induced current in the
circuit.
 
 B  B  S  BS cos 0  BS
d B d ( BS ) dB


S  (0.020 T / s)(0.012 m 2 )  2.4 10  4 V  0.24mV
dt
dt
dt
EMF
2.4 104V
i

 4.8 105 A  0.048mA
R
5.0
A Simple Generator
Induced EMF and Inductance
1830s Michael Faraday
Joseph Henry
B  i;   B;   Mi
M is mutual inductance