### Topic 12.1 Induced electromotive force (emf)

```Topic 12.1
Induced electromotive force (emf)
3 hours
Electromagnetic Induction
• Electromagnetic induction is the production
of electric current across a conductor moving
through a magnetic field. It underlies the
operation of generators, transformers,
induction motors, electric motors,
synchronous motors, and solenoids.
• Michael Faraday is generally credited with
the discovery of induction in 1831 though it
may have been anticipated by the work of
Francesco Zantedeschi in 1829. Around the
same time Joseph Henry made a similar
discovery, but did not publish his findings
until later.
Induced Current and emf
• Consider a wire of length l that is moved with
velocity v in a region of magnetic field B.
• When the wire conductor moves in the magnetic field, the free
electrons experience a force because they are caused to move with
velocity v as the conductor moves in the field.
F=evB
• This force causes the electrons to drift from one end of the
conductor to the other, and one end builds-up an excess of
electrons and the other a deficiency of electrons. This means that
there is a potential difference or emf between the ends.
• Eventually, the emf becomes large enough to balance the magnetic
force and thus stop electrons from moving.
e v B = e E ⇔E = B v
• If the potential difference (emf) between the ends of the conductor
is ε then
ε=El
• By substitution, we have,
ε=Bvl
• If the conducting wire was a tightly wound coil of N turns of wire
the equation becomes:
ε=NBvl
(Students should be able to derive the expression induced emf = Blv without using Faraday’s law.)
Magnetic Flux
• Consider a small planar coil of
a conductor for simplicity as
shown (it could be any small
shape).
• Now imagine it is cut by
magnetic lines of flux. It would
be reasonable to deduce that
the number of lines per unit
cross-sectional area is equal to
the magnitude of the
magnetic flux density B × the
cross-sectional area A. This
product is the magnetic flux
F.
Magnetic Flux
• The magnetic flux Φ through a small plane
surface is the product of the flux density normal
to the surface and the area of the surface.
F=BA
• The unit of magnetic flux is the weber Wb.
Rearranging this equation it can be seen that:
B = F / A which helps us understand why B can
be called the flux density. So the unit for flux
density can be the tesla T, or the weber per
square metre Wb m-2. So, 1T = 1 Wb m-2
• If the normal, shown by the dotted line in the
figure, to the area makes an angle θ with B, then
the magnetic flux is given by:
F = B A cosq
• where A is the area of the region and q is the
angle of movement between the magnetic field
and a line drawn perpendicular to the area swept
out.
(Be careful that you choose the correct vector
component and angle because questions on past
IB examinations give the correct answers of BA sin
q or BA cos q depending on components supplied
in the diagrams).
• If F is the flux density through a cross-sectional
area of a conductor with N coils, the total flux
density will be given by:
F = N B A cosq
• This is called flux linkage.
• So it should now be obvious that we can increase
the magnetic flux by:
• Increasing the conductor area
• Increasing the magnetic flux density B
• Keeping the flux density normal to the surface
of the conductor
Changing Flux
• The figure below shows the shaded area of
the magnetic flux density swept out in one
second by a conductor of length l moving from
the top to the bottom of the figure through a
distance d.
• We have already derived that ε = B l v
• The area swept out in a given time is given by (l × d) / t.
• But v = d / t. So that the area swept out per unit time is
given by A/Dt = lv. That is,
where A is the area in m2.
• For a single conductor in the magnetic flux density, it
can be seen that:
• where the constant equals –1. The negative sign
indicates that direction of the induced emf is such that
the current it causes to flow opposes the change
producing it.
Faraday’s Law and Lenz’s Law
• If there are N number of coils, then:
• Faraday’s Law can therefore be stated as: The
magnitude of the induced emf in a circuit is
directly proportional to rate of change of
magnetic flux or fluxlinkage.
• Lenz’s Law gives us the minus sign and can be
stated as: The direction of the induced emf is
such that the current it causes to flow
opposes the change producing it.
Lenz’s Law applied to a solenoid.
Lenz’s Law Applied to a straight conductor.
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