PES 1120 Spring 2014, Spendier
Lecture 32/Page 1
Today: start chapter 30
- Induction
- Magnetic flux
- Faraday’s Law: Induced emf
So far we have seen:
- electric current ==> magnetic field (chapter 29)
- current through a wire ( interacting B-fields ==> force (or torque) on the wire
(chapter 28)
In this chapter we will look at
- moving wire + magnetic field ==> induced current in the wire (chapter 30)
Question: Electricity creates Magnetism. Can Magnetism create Electricity?
Induction Experiments
PHET: http://phet.colorado.edu/en/simulation/faraday
It wasn’t long after people started looking at electric charge and current that it was
noticed that magnetic fields could also cause a current. In 1831, Michael Faraday
discovered that, by varying magnetic field with time, an electric field could be generated. The
phenomenon is known as electromagnetic induction. The Figure below illustrates one of
Faraday’s experiments.
Faraday showed that no current is registered in the galvanometer when bar magnet is
stationary with respect to the loop. However, a current is induced in the loop when a relative
motion exists between the bar magnet and the loop. In particular, the galvanometer deflects in
one direction as the magnet approaches the loop, and the opposite direction as it moves away.
Faraday’s experiment demonstrates that an electric current is induced in the loop by changing
the magnetic field. The coil behaves as if it were connected to an emf source. Experimentally
it is found that the induced emf depends on the rate of change of magnetic flux through the
coil. The faster we do any of these things, the greater the induced emf.
An emf is induced in the loop when the number of magnetic field lines that pass
through the loop is changing.
PES 1120 Spring 2014, Spendier
Lecture 32/Page 2
Hence the induced emf depends on the change in the amount of magnetic field that passes
through the loop. We will see that there is indeed a relationship. But first we need to talk
about magnetic flux in more detail.
Magnetic flux
To calculate the induced emf, we need a way to calculate the amount of magnetic field
that passes through a loop. In Chapter 23, in a similar situation, we needed to calculate
the amount of electric field that passes through a surface. There we defined an
electric flux  E . Here we define a magnetic flux  B : Suppose a loop enclosing
an area A is placed in a magnetic field .Then the magnetic flux through the loop is
 
 B   B  dA   B dA cos( )
units: [Wb] = [T m2]
weber (Wb)
NOTE: To avoid getting zero, we do NOT find the
total flux through the loop. We only do the points on

the loop where
the
area
vectors
are
parallel
to
B
 
 
 B   B  dA instead of  B  dA

As in Chapter 23, dA is a vector of magnitude dA that is perpendicular to a differential
area dA
Example: For a square loop of wire with side length L, what is the flux through the loop
in each of these cases:
PES 1120 Spring 2014, Spendier
Lecture 32/Page 3
Faraday’s Law
The first physicist (that we know of) to study magnetic induction was Michael Faraday in
the 1800’s. We can encompass all of these experimental facts into one equation. The
induced emf , εind, in a coil is proportional to the negative of the rate of change of magnetic
flux:
 
dB
d
d

B
 dA  
B dA cos( )

dt
dt
dt 
emf induced = rate of change in magnetic flux
 ind  




Faraday’s Law: The rate at which the magnetic flux coming out of a conducting loop
changes determines the induced EMF, εind. The negative sign tells us that the direction of
the emf and resulting current is such as to oppose the change in magnetic field. This is
known as Lenz's Law which we will discuss in more detail below.
For a coil that consists of N loops, the total induced emf would be N times as large:
dB
 ind   N
dt
Magnetic Induction - The creation of an electric field from a changing magnetic
field.
Different ways that an emf may be induced
 
 B   B  dA   B dA cos( )
For a loop of area A, we can write:
 ind  
d
dB
dA
d
 BA cos       A cos   B   cos  BA sin  
dt
 dt 
 dt 
 dt 
Thus, we see that an emf may be induced in the following ways:

(i) by varying the magnitude of B with time
PES 1120 Spring 2014, Spendier
Lecture 32/Page 4

(ii) by varying the magnitude of A , i.e., the area enclosed by the loop with time, i.e inducing
emf by changing the area of the loop.


(iii) varying the angle between B and the area vector A with time
Lenz’s Law
The direction of the induced current is determined by
Lenz’s law: The induced magnetic field opposes the change in flux that created it
There are two magnetic fields in every induction problem!

B0 - The original field from a bar magnet or some other source

Changing B0 Flux ==> Induced Current

Bind - The induced field created by the induced current
PES 1120 Spring 2014, Spendier
Lecture 32/Page 5
Lenz’s Law has two versions depending on whether the original field’s flux is
increasing or decreasing
Increasing Flux:


The induced magnetic field Bind tries to cancel B0 .
Decreasing Flux:


The induced magnetic field Bind tries to maintain B0 .
Example
If we move a bar magnet towards and away from a wire loop, we can plot the flux in the
loop and the induced emf (proportional to Iind) in the loop as a function of time
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Lecture 32/Page 6
Example: A loop of wire with radius R = 2.5 cm is moved from outside a uniform field
to inside the uniform field (B=0.056 T) in a time of 0.30s.
 
a) What is the induced emf? Assume B  A (out of the page)
b) Which way will the induced current flow?
PES 1120 Spring 2014, Spendier
Lecture 32/Page 7
Extra Example - not covered in class
A 120-turn coil of radius 1.8 cm and resistance 5.3  is coaxial with a solenoid of 220
turns/cm and diameter 3.2 cm. The solenoid current drops from 1.5 A to zero in time
interval ∆t = 25 ms. What current is induced in the coil during ∆t?