1E6 Electrical Engineering
Electricity and Magnetism
Lecture 19: Electromagnetic Induction
19.1 Magnetic Fields and Current Carrying Conductors
When a conductor which is carrying an electric current is placed in a
magnetic field, the field generated around the current carrying conductor
interacts with the magnetic field into which it is placed.
Consider the conductor shown in Fig. 1, which is placed into the magnetic
field created by two permanent magnets with opposite poles facing each other.
In this case the conductor has no current flowing through it. It simply resides in
the magnetic field and the magnetic flux generated between the North and South
poles of the magnets passes through the conductor which has little influence on
this field.
permanent
magnet
permanent
magnet
S
N
conductor
carrying
no current
Fig. 1.
A Conductor Carrying no Current Placed in a Magnetic Field
Now consider passing a current, I, through the conductor. The current
flowing through the conductor generates a magnetic field around the conductor
as shown in Fig. 2. The direction of this magnetic field is given by the Right
Hand Screw Rule as before. In the case of current flowing away from an
observer of the page, the magnetic field associated with the conductor will have
a clockwise direction as shown. The magnetic fields generated by the permanent
magnets and the current flowing in the conductor interact with each other.
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The interaction between the two fields is based on the rules which apply between
lines of flux. In this case, parallel lines of flux running in the same direction tend
to repel each other while lines running in opposite directions tend to attract each
other. As can be seen in Fig. 2, the lines of flux in the magnetic field created by
the permanent magnets in the region above the conductor run in the same
direction as those of the field generated by the current flowing in the conductor.
Consequently, in the region above the conductor the two fields tend to repel each
other. If the permanent magnets are considered rigid but the conductor mobile,
then this gives rise to a force tending to move the conductor downward. On the
other hand, in the region underneath the conductor the lines of flux of the two
fields run in opposite directions and therefore the two fields tend to attract each
other. In this case there is a consequent attractive force on the conductor
tending to pull it downward. The overall effect of the interaction of the two fields
is that there is a net force acting in the downward direction on the conductor,
and at a right angle to it. This is a physical force which will cause the conductor
to move if it is supported in such a way that it is free to do so.
permanent
magnet
S
N
permanent
magnet
conductor
carrying a
current
downward force
experienced by
the conductor
Fig. 2.
A Conductor Carrying a Current Placed in a Magnetic Field
Finally, if either the direction of the permanent magnetic field or the
direction of the current flowing in the conductor is reversed, then the direction
of the force acting on the conductor is also reversed.
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19.2 Magnetic Flux and Force
In the above case of the current carrying conductor placed in the magnetic
field, it is useful to be able to establish the directional relationship between the
three parameters involved namely: the magnetic field associated with the
permanent magnets, the current flowing in the conductor and the resulting force
acting on the conductor. This can be done by a useful rule known as the Left
Hand Rule. This rule is sometimes attributed to John Ambrose Fleming (1849 –
1945), a British physicist and electrical engineer, but in fact was not formally
stated by him.
(Fleming’s) Left Hand Rule is as follows:
Holding the left hand out with the first finger, second finger and thumb all at
mutual right angles to each other as shown in Fig. 3:
The First finger indicates the direction of the magnetic Field of the magnet
The SeCond finger indicates the direction of the Current
The ThuMb indicates the direction of Motion of the conductor.
Fig. 3 An Illustration of (Fleming’s) Left Hand Rule.
By aligning any two of the digits in the directions of their respective associated
parameters, the direction of the third can be found. Commonly, this rule is used
to find the direction of movement of the conducting windings in a motor.
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The resulting force acting on a current carrying conductor placed in a magnetic
field is proportional to the intensity of the magnetic field and the magnitude of
the current flowing in the conductor. It is also dependent on the length of the
conductor which is exposed to the magnetic field.
Force is given the symbol F and has units of Newtons (N)
Then:
Force = Magnetic Flux Density x Current x Length of Conductor
F= BIl
N
Case Study 1:
The faces of the poles of a permanent magnet have dimensions 150mm
high x 250mm wide. A total magnetic flux of 0.5 Wb exists between the poles of
the magnet. Determine the force acting on a conductor carrying a current of
200mA which runs horizontally between the poles of the magnet. Evaluate the
acceleration experienced by the conductor if it has a mass of 100g/m and is free
to move within this field.
Solution:
A = 0.15 x 0.25 m2 = 0.0375 m2
250mm
150mm
conductor
B = Ф/A =0.5 / 0.0375 = 13.3 T
N
S
The length of the conductor l which is
exposed to the magnetic field is the
width of the face of the magnet of
250mm.
F = BIl = 13.3 x 0.2 x .25 = 0.665 N
From Newton’s laws of motion we have
Force = Mass x Acceleration,
F = ma.
where m is the mass of 250mm length of conductor m = 0.1 x .25 = 0.025 kg
Then acceleration a = F / m = 0.665 / 0.025 = 26.6 m/s2
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19.3 Electromagnetic Induction
The interactions of a conductor in a magnetic field can be reversed. That
is, the flux associated with the magnetic field can be moved and an electric force
induced in the conductor which, if a closed path is provided, will cause current
to flow in the conductor. Fig. 4 shows such a scenario. A magnetic flux exists
between the poles of the magnets as before. A conductor is mounted vertically
between the poles of the magnets and the magnet is moved sideways as viewed in
the plan, such that the magnetic flux between the poles cuts the conductor which
is immobile. This is equivalent to the conductor moving in the opposite relative
direction through the magnetic field. The magnetic flux cutting the conductor
exerts an influence on the electrons in the conductor and generates an electrical
force tending to cause them to move, i.e. a force which tends to induce current
flow in the conductor. This force is known as an Electromotive Force or EMF.
This EMF is only generated while the magnetic flux cuts the conductor and
hence only while the magnet is moving. The polarity of the EMF generated and
any current which flows through a load connected across the terminals of the
conductor will be reversed if the direction of motion of the magnet is reversed.
direction
of induced
current
relative direction of
motion of conductor
permanent
magnet
permanent
magnet
conductor
S
N
S
N
conductor
permanent
magnet
permanent
magnet
direction of
motion of magnet
Plan
Elevation
+
Fig. 4 Magnetic Flux Cutting a Conductor to Generate an EMF
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19.4 Magnetic Flux and Induced EMF
In the case of electromagnetic induction it is again useful to be able to
establish the directional relationship between the three parameters involved
namely: the magnetic field associated with the permanent magnets, the direction
of motion of the magnets and the induced emf (and hence the current flowing) in
the conductor. This can be done by the use of Fleming’s Right Hand Rule.
Fleming’s Right Hand Rule is as follows:
Holding the right hand out with the first finger, second finger and thumb all at
mutual right angles to each other as shown in Fig. 5:
The First finger indicates the direction of the magnetic Field of the magnet.
The ThuMb indicates the direction of Motion of the conductor relative to
the magnetic field.
The SECond finger indicates the direction of the induced EMF or Current
Fig. 5 An Illustration of Fleming’s Right Hand Rule.
By aligning any two of the digits in the directions of their respective associated
parameters, the direction of the third can be found. Commonly, this rule is used
to find the direction of the emf produced by a generator. Another important rule
associated with electromagnetic induction is Lenz’s Law
Lenz’s Law states that: the direction of an induced emf is always such that it tends
to set up a current opposing the motion or the change of flux responsible for
inducing that emf.
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The resulting emf generated when magnetic flux cuts a conductor is
proportional to the intensity of the magnetic field, the length of the conductor
which is cut by the flux and the rate at which the flux cuts the conductor.
EMF is given the symbol E and has units of Volts (V)
Then:
EMF = Magnetic Flux Density x Length of Conductor x Relative Velocity
E = Bl u
V
If the magnets travel a distance, d, in T seconds, then the rate at which the flux
cuts the conductor is:
u=
d
T
Which gives:
E=
Bld
T
But the total area of the flux which cuts the conductor in the time, T, is the
length of the conductor, l, within the magnetic field times the distance travelled,
d, so that:
A = ld
BA = Φ
and since:
then:
E=
BA Φ
=
T
T
V (or Wb/s)
On an instantaneous basis the magnetic flux cutting the conductor could change
itself. It is still, however, the rate at which the flux cuts the conductor which
determines the induced instantaneous emf, e.
In this case:
e(t) =
dΦ
dt
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V
Case Study 2:
The faces of the poles of a permanent magnet have dimensions 150mm
high x 250mm wide. A total magnetic flux of 0.5 Wb exists between the poles of
the magnet. The magnet moves from left to right at a speed of 60cm/s.
Determine the emf induced in a conductor mounted vertically between the faces
of the magnet while within the magnetic field and the relative polarity of this
emf.
Solution:
A = 0.15 x 0.25 m2 = 0.0375 m2
150mm
250 mm
B = Ф/A =0.5 / 0.0375 = 13.3 T
N
as before.
motion of
magnet
The length of the conductor l which is
exposed to the magnetic field in this
case is the height of the face of the
magnet of 150mm.
S
relative
motion of
conductor
The relative velocity of the conductor is u = 60 cm/s = 0.6 m/s
Then:
e = Blu = 13.3 x 0.15 x .6 = 1.197 ≈ 1.2 V
Using Fleming’s Right Hand Rule,
noting that the relative direction of
motion of the conductor is the opposite
to that of the magnet, i.e. from right to
left, it can be established that the emf is
induced in the conductor acting in an
upwards direction from bottom to top.
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