Electromagnetic Induction
Finally! Flux!
Objectives
 Calculate flux or flux linkage using 𝜙 = 𝐵𝐴 cos 𝜃 or 𝜙 =
𝑁𝐵𝐴 cos 𝜃;
 Identify situations in which an emf is induced and determine the
magnitude of the emf using Faraday’s law, 𝜀 =
𝑑𝜙
; included
dt
∆𝜙
or
∆t
𝜀=
−
are cases of changing area, a changing
magnetic field, or a changing angle between magnetic field and
normal to the loop;
 Find the direction of the induced current using Lenz’s law.
A wire moving in a magnetic field
Imagine a wire of length L is moved with velocity v in a
region of a magnetic field of constant magnitude B.
Assume for convenience that B is coming out of the
page and the wire moves from top to bottom.
The wire is conducting
(meaning it has many
free electrons.
A wire moving in a magnetic field
As the wire moves, the electrons also move from top to
bottom. Thus the magnetic field will exert a force on these
moving electrons (force on a moving charge, not current in
a wire).
The force on the
electrons is directed
from left to right and
therefore the electrons
are pushed to the right.
A wire moving in a magnetic field
This means that the left end of the wire has a net positive charge
and the right end of the wire has an equal net negative charge
(the net charge is still zero). This establishes an electric field in
the wire whose direction is from left to right. The magnitude of
∆𝑉
∆𝑥
𝑉
; where V
𝐿
this field is 𝐸 = =
is the potential difference
between the ends of the wire that is established because of the
accumulation of charge at its ends.
A wire moving in a magnetic field
• The flow of electrons will thus stop when the electric
force, eE, equals the magnetic force, evB, pushing them
towards the right. Thus
𝑉
𝑒
𝐿
𝑒𝐸 = 𝑒𝑣𝐵 →
= 𝑒𝑣𝐵 ∴ 𝑉 = 𝑣𝐵𝐿.
• We have found the extraordinary result that a conducting
wire of length L moving with speed v normally to a
magnetic field B will have a potential difference of vBL
across its ends. This is called a motional emf: it has been
induced as a result of the motion of the conductor in the
magnetic field. (If you don’t believe me, check the units. It
works!)
Lastly
 It is no longer the case that the electric field inside a
conductor is zero when charges are allowed to move
(that was static electricity only).
Faraday’s Law
 As we saw earlier, an electric current creates a
magnetic field.
 In the previous section we saw that a wire that
moves in a magnetic field has an induced emf
at its ends.
 Actually producing a current by a magnetic field was
a difficult problem in nineteenth-century physics.
Faraday’s Law - Experiment
 Consider the following experiment.
 A magnet is moved towards a loop
of wire whose ends are connected
to a sensitive galvanometer and in a
direction normal to the plane of the
loop as shown.
 The galvanometer registers a
current.
Faraday’s Law – Experiment
observations
 If the magnet is simply placed near the coil,
but does not move relative to it, nothing
happens.
 The current has been created as a result of
the motion of the magnet relative to the loop
of wire.
 If we now move the magnet toward the coil
faster, the reading on the galvanometer is
greater.
 If we move the coil toward the magnet, we
again find a reading.
Faraday’s Law – Experiment
implications
• This indicates that it is the relative motion of the coil and
magnet that is responsible for the effect.
• If a stronger magnet is used, the effect is greater.
 If we use a different loop of wire with the
same area but more turns, the effect is
greater.
 If the area of the single loop is increased, the
current also increases.
 If the magnet is moved at an angle other than
90°, the current decreases.
Faraday’s law – Experiment summary
The observations are that the current registered by the
galvanometer increases when:
 The relative speed of the magnet with respect to the coil
increases;
 The strength of the magnet increases;
 The number of turns increases;
 The area of the loop increases;
 The magnet moves at a right angle to the plane of the
loop.
Faraday’s law – Magnetic Flux
 Faraday found that the common thread behind all these
observations is the concept of magnetic flux. Imagine a
loop of wire in a magnetic field whose magnitude and
direction is constant, then we define flux as follows.
Faraday’s law – Magnetic Flux
 The magnetic flux φ through the loop is 𝜙 = 𝐵𝐴 cos 𝜃;
where A is the area of the loop, and θ is the angle
between the magnetic field direction and the direction
normal to the loop area.
 If the loop has N turns of wire around it, the flux is given
by 𝜙 = 𝑁𝐵𝐴 cos 𝜃; in which case we speak of flux
linkage.
 The unit of magnetic flux is the Weber (Wb): 1Wb =
1Tm2.
Faraday’s law – Magnetic Flux
 This means that if the magnetic field is along the
plane of the loop, then θ=90° and hence φ= 0. The
maximum flux through the loop occurs when θ =
0°, when the magnetic field is parallel to the loop
area and its value is then BA.
Minimum flux
Maximum flux
Faraday’s law – Magnetic Flux
 The intuitive picture of magnetic flux is the number
of magnetic field lines that cross or pierce the
loop area.
 Note that if the magnetic field went through only half
the loop area, the other half being in a region of no
magnetic field, then the flux in that case would be 𝜙 =
𝐵𝐴
.
2
 In other words, what counts is the part of the loop area
that pierced by the magnetic field lines.
Faraday’s law – Magnetic Flux
Thus, to increase the magnetic flux of a loop of wire we must:
 Increase the loop area that is exposed to the magnetic field;
 Increase the value of the magnetic field;
 Have the loop normal to the magnetic field.
Example
A loop of area 2 cm2 is in a constant
magnetic field of B = 0.10 T. What is
the magnetic flux through the loop
when:
 The loop is perpendicular to the
field;
 The loop is parallel to the field;
 The normal to the loop and the
field have an angle of 60° between
them?
Faraday’s law - Magnetic Flux
 What does magnetic flux have to do with the problem of
how a magnetic field can create an electric field? The answer
lies in a changing magnetic flux.
 In all the cases we described, we had a magnetic flux
through the loop, which was changing with time. As a
magnet is brought closer to the loop area, the value of the
magnetic field at the loop portion is increasing and so is
flux.
 If the magnet is held stationary near the loop, there is flux
through the loop, but it isn’t changing – so nothing
happened. If the number of turns is increased, so is the flux
linkage.
Faraday’s law - Magnetic Flux
 Thus, there seems to be a connection between the
amount of current induced, and the rate of change of
magnetic flux linkage through the loop. This is known as
Faraday’s law.
 The induced emf is equal to the negative rate of
change of magnetic flux: 𝜺 =
∆𝝓
−𝑵 .
∆𝐭
 (The minus sign isn’t important until we use calculus, so
you can forget it for now since we are finding the
magnitude.)
Example
The magnetic field through a single loop of area 0.2 m2
is changing at a rate of 4 T/s. What is the induced emf?
Example
A uniform magnetic field B = 0.40 T is established into
the page. A rod of length L = 0.20 m is placed on a
railing and pushed to the right with constant speed v =
0.60 m/s. What is the induced emf in the loop?
Lenz’s Law
 Thinking about the last example, we need to ask the
question, which direction does the current flow?
 There are two options. If the current flows counterclockwise, by the right hand rule, force is directed right – in
the direction of motion of the rod. Or, if the current flows
clockwise, the force is directed to the left – oposing te
motion of the rod.
 Which makes physical sense?
Lenz’s Law
 If current flows counter-clockwise, then the rod is
accelerated to the right.
 An increased speed leads to increased emf, which
leads to increased current and so on and so on
forever.
 But with no one providing the necessary energy, this
scenario violates the conservation of energy.
Lenz’s Law
 Clearly, the induced current must act to
oppose the change in magnetic flux that
created the current.
 This is Lenz’s law.
Lenz’s Law
 This is a little tricky to think about, so lets stick with
the scenario in the last example.
 The change in the magnetic flux was a decrease
because the area got smaller. The induced current
will create its own magnetic field (Ørsted’s
discovery), which will also have a flux through the
same loop.
Lenz’s Law
 If the created magnetic field is in the same direction
as the original magnetic field, its flux will add to the
original (which was decreasing) and prevent it from
decreasing as fast.
 In this case, the induced current must flow in a
clockwise direction, as we found earlier!
Example
A loop of wire has its plane horizontal and a bar
magnet is dropped from above so that it falls through
the loop with (a) the north pole first and (b) the south
pole first. Find the direction of the current induced in
the loop in each case.
Summary
 State the formula to calculate Flux
 When is an emf induced? How can we calculate its
magnitude?
 How can we determine the direction of the induced current
in a wire?