Induced Voltages and
Inductances
Induced Emf and Magnetic Flux
 Discovery of induction
Induced Emf and Magnetic Flux

Magnetic flux
A (area)
magnetic flux:  B  BA

B

A  Anˆ
n̂ : Normal unit vector
Unit vector perpendicular to the plane

A  Anˆ
 
magnetic flux:  B  BA cos q  B A  B  A
q

B
Farady’s Law of Induction
 Farady’s
law of induction
An emf in volts is induced in a circuit that is equal to the time rate of
change of the total magnetic flux in webers threading (linking) the
circuit:
 B
 
t
 B 

   N t 
If the circuit contains
N tightly wound loops
The flux through the circuit may be changed in several different ways
1) B may be made more intense.
2) The coil may be enlarged.
3) The coil may be moved into a region of stronger field.
4) The angle between the plane of the coil and B may change.
Farady’s Law of Induction
 Farady’s
law of induction (cont’d)
•The sum of Ei si along the loop
is equal to the work done per unit
charge, which is the emf of the circuit.
  
   Ei  sdi s  Ei si cos qi
i
i
Farady’s Law of Induction

Lenz’s law
The sign of the induced emf is such that it tries to produce a
current that would create a magnetic flux to cancel (oppose) the
original flux change.
B due to induced current
B due to induced current
Farady’s Law of Induction

Lenz’s law (cont’d)
• The bar magnet moves towards loop.
• The flux through loop increases, and an emf induced in the loop
produces current in the direction shown.
• B field due to induced current in the loop (indicated by the dashed
lines) produces a flux opposing the increasing flux through the loop
due to the motion of the magnet.
Motional Electromotive Force
 Origin
of motional electromotive force I
FE
FB
Motional Electromotive Force
 Origin
of motional electromotive force I (cont’d)
Motional Electromotive Force
 Origin
of motional electromotive force II
.
B
Bind
Motional Electromotive Force
 Origin
of motional electromotive force II (cont’d)
 B
 
t
Motional Electromotive Force
 Origin
of motional electromotive force II (cont’d)
 
 B
Blx
x

  Bl
  Blv
t
t
t
Motional Electromotive Force
 Origin
of motional electromotive force II (cont’d)
Motional Electromotive Force
 Origin
of motional electromotive force II (cont’d)
Motional Electromotive Force
 Origin
of motional electromotive force III
Motional Electromotive Force
 Origin
of motional electromotive force III (cont’d)
F  ma  m lim t 0
v
t
B 2 2v
v
dv

 m lim
m

t

0
R
t
dt
Motional Electromotive Force
 Origin
of motional electromotive force III (cont’d)
JUST FOR FUN WITH CALCULUS!
B 2 2v
v
dv

 m lim
m
t 0 t
R
dt
Motional Electromotive Force
 A bar
magnet and a loop (again)
In this example, a magnet is being pushed towards (away from) a closed loop.
The number of field lines linking the loop is evidently increasing (decreasing).
Motional Electromotive Force
 An
electromagnet and a coil
Motional Electromotive Force
 Tape
recorder
Generators

v
A generator (alternator)
The armature of the generator
opposite is rotating in a uniform B field
with angular velocity ω this can be
treated as a simple case of the E =
υ×B field.
On the ends of the loop υ×B is
perpendicular to the conductor so
does not contribute to the emf. On the
top υ×B is parallel to the conductor
and has the value E = υB sin θ =
ωRB sin ωt. The bottom conductor
has the same value of E in the
opposite direction but the same sense
of circulation.
q
top
90oq
B
bottom
 
AB   Ei  si  2LRB sin t  AB sin t
i
v
Farady emf
Generators

A generator (cont’d)
AC generator
DC generator
induced
current
Self-inductance
 Self-inductance
Consider the loop at the right.
X XX X
X XX XX X
X XX X
- Switch closed : Current starts to flow on the loop. a
b
switch
- Magnetic field produced in the area enclosed by
the loop (B proportional to I).
- Flux through the loop increases with I.
- Emf induced to oppose the initial direction of the current flow.
- Self-induction: changing the current through the loop inducing
an opposing emf the loop.
Self-inductance
 Self-inductance
(cont’d)
- The magnetic field induced by the current
in the loop is proportional to the current:
I
BI
- The magnetic flux induced by the current
in the loop is also proportional to the current:
- Define the constant of proportion as L:
- From Frarady’s law:
SI unit of L :
 B
I
L  
 L
t
t
 
 B   Bi  si  I
B
L
I
Wb
T m2
1H 1
1
A
A
i
self-inductance
Self Inductance
 Calculation
of self inductance : A solenoid
Accurate calculations of L are generally difficult. Often the answer
depends even on the thickness of the wire, since B becomes strong close
to a wire.
In the important case of the solenoid,
the first approximation result for L is
quite easy to obtain: earlier we had
N
B  0 I

Then,
Hence
N2A
 B  NAB  0
I

B
N2A
L
 0
 0 n 2 A
I

n : the number of turns
per unit length
So L is proportional to n2 and the volume of the solenoid
Self Inductance
 Calculation
of self inductance: A solenoid (cont’d)
N2A
L  0
 0 n 2 A

n : the number of turns
per unit length
Example: the L of a solenoid of length 10 cm, area 5 cm2, with a total
of 100 turns is
L = 6.28×10−5 H
0.5 mm diameter wire would achieve 100 turns in a single layer.
Going to 10 layers would increase L by a factor of 100. Adding an iron
or ferrite core would also increase L by about a factor of 100.
The expression for L shows that μ0 has units H/m, c.f, Tm/A obtained earlier
RL Circuits
 Inductor
A circuit element that has a large inductance, such as a closely
wrapped coil of many turns, is called a inductor.

RL circuit
Kirchhoff’s rules:
 IR  L  0
I
dI
L   L
 L   L
t
dt
dI
 IR  L  0
dt


 Rt / L
I  (1  e
)  (1  e t / )
R
R
time constant
Energy Stored in a Magnetic Field
 Inductor
• The emf induced by an inductor prevents a battery from establishing
instantaneous current in a circuit.
• The battery has to do work to produce a current – this work can be
considered as energy stored in the inductor in its magnetic field.
PE L 
1 2
LI
2
energy stored in inductor