#### Theory of Electromagnetic Induction, Magnetic Fields and Current Carrying Conductors

Magnetic Fields and Current Carrying Conductors:

When a conductor that is carrying an electric current is positioned in a magnetic field, the field formed around the current carrying conductor interacts with the magnetic field into which it is positioned.

Consider the conductor as shown in figure below that is placed into the magnetic field formed by two permanent magnets with opposite poles facing one other. In this case the conductor has no current flowing via it. It simply resides in magnetic field and the magnetic flux generated between the North and South poles of the magnets passes via the conductor that has little influence on this field.

Figure: A Conductor carrying no current positioned in a Magnetic field

Now consider passing a current, I, via the conductor. The current flowing via the conductor generates a magnetic field about the conductor as shown in figure below. The direction of such magnetic field is given by the Right Hand Screw Rule. In the case of current flowing away from an observer of the page, the magnetic field related with the conductor will have a clockwise direction as shown. The magnetic fields produced by the permanent magnets and the current flowing in the conductor interact with one other.

The interaction among the two fields is based on the rules that apply between lines of flux. In this case, parallel lines of flux running in similar direction tend to repel each other whereas lines running in opposite directions tend to attract one other. As can be seen in figure below, the lines of flux in the magnetic field made by the permanent magnets in the area above the conductor run in the similar direction as those of the field produced by the current flowing in the conductor. Consequently, in the area above the conductor the two fields tend to repel one other. When the permanent magnets are considered rigid however the conductor mobile, then this gives mount to a force tending to move the conductor downward. On other hand, in the area underneath the conductor, the lines of flux of two fields run in opposite directions and thus the two fields tend to attract one other. In this case there is a subsequent attractive force on the conductor tending to pull it downward. The total effect of interaction of the two fields is that there is total force acting in the downward direction on the conductor, and at a right angle to it. It is a physical force that will cause the conductor to move if it is supported in such a manner that it is free to do so.

Figure: A Conductor Carrying a Current Placed in a Magnetic Field

Ultimately, if either the direction of permanent magnetic field or the direction of current flowing in the conductor is reversed, then the direction of the force acting on conductor is as well reversed.

Magnetic Flux and Force:

In above case the current carrying conductor positioned in the magnetic field, is useful to be able to establish the directional relationship among the three parameters comprised namely: the magnetic field related with the permanent magnets, the current flowing in the conductor and the resultant force acting on the conductor. This can be done by a helpful rule termed as the Left Hand Rule. This rule is sometimes credited to John Ambrose Fleming (1849 – 1945), a British physicist and electrical engineer, however in fact was not formally defined by him.

Fleming’s Left Hand rule is as follows:

Holding the left hand out with first finger, second finger and thumb all at mutual right angles to one other as shown in figure below:

•    The First finger points out the direction of magnetic Field of the magnet
•    The Second finger points out the direction of Current
•    The Thumb points out the direction of Motion of the conductor.

Figure: An Illustration of Fleming’s Left Hand Rule.

By aligning any two of digits in the directions of their respective related parameters, the direction of the third can be found. Generally, this rule is employed to find the direction of movement of the conducting windings in the motor.

The resultant force acting on a current carrying conductor positioned in a magnetic field is proportional to the intensity of magnetic field and the magnitude of the current flowing in the conductor. This is as well dependent on the length of conductor that is exposed to the magnetic field.

Force is represented by the symbol F and the unit is Newtons (N)

Then,

Force = Magnetic flux density x Current x Length of conductor

F = B I l N

Illustration: The faces of poles of a permanent magnet contain dimensions 150mm high x 250mm wide. The total magnetic flux of 0.5 Wb exists among the poles of magnet. Find out the force acting on a conductor carrying a current of 200mA that runs horizontally among the poles of magnet. Calculate the acceleration experienced by the conductor if it consists of a mass of 100g/m and is free to move in this field.

Solution:

A = 0.15 x 0.25 m2 = 0.0375 m2

B = Φ/A =0.5 / 0.0375 = 13.3 T

The length of conductor l that is exposed to the magnetic field is the width of 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 encompass, 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

Electromagnetic Induction:

The interactions of conductor in a magnetic field can be reversed. That is, the flux related with the magnetic field can be moved and an electric force induced in the conductor that, if a closed path is given, will cause current to flow in the conductor. Figure below shows such a scenario. The magnetic flux exists between the poles of the magnets as prior to. The conductor is mounted vertically among the poles of magnets and the magnet is moved sideways as viewed in the plan, such that the magnetic flux among the poles cuts the conductor that is immobile. This is equivalent to the conductor moving in the opposite relative direction via the magnetic field. The magnetic flux cutting the conductor exerts an effect on the electrons in the conductor and produces an electrical force tending to cause them to move, that is, a force that tends to induce current flow in the conductor. This force is termed as an Electromotive Force or EMF. This EMF is only produced while the magnetic flux cuts the conductor and therefore only while the magnet is moving. The polarity of the EMF produced and any current that flows via a load connected across the terminals of the conductor will be reversed when the direction of motion of the magnet is reversed.

Figure: Magnetic Flux Cutting a Conductor to Generate an EMF

Magnetic Flux and Induced EMF:

In case of electromagnetic induction it is again helpful to be able to establish the directional relationship among the three parameters comprised namely: the magnetic field related with permanent magnets and direction of motion of the magnets and the induced emf (and hence the current flowing) in the conductor. It can be done by the utilization of Fleming’s Right Hand Rule.

Fleming’s Right Hand Rule is as follows:

Holding the right hand out with first finger, second finger and thumb all at mutual right angles to one other as shown in figure below:

•    The First finger points out the direction of the magnetic Field of magnet.
•    The Thumb points out the direction of Motion of the conductor associative to the magnetic field.
•    The Second finger points out the direction of the induced EMF or Current.

Figure: An Illustration of Fleming’s Right Hand Rule.

By aligning any two digits in the directions of their respective related parameters, the direction of the third can be found. Generally, this rule is employed to find the direction of the emf generated by a generator. The other important rule related with electromagnetic induction is Lenz’s Law

Lenz’s Law defines that: “the direction of an induced emf is always such that it tends to set up a current opposing the motion or change of flux accountable for inducing that emf”.

The resultant emf generated whenever magnetic flux cuts a conductor is proportional to the intensity of magnetic field, the length of conductor that is cut by the flux and the rate at which the flux cuts the conductor.

EMF is represented by the symbol E and consists of units of Volts (V)

Then,

EMF = Magnetic flux density x Length of conductor x Relative velocity

E = B l u V

If the magnets travel a distance d, in T seconds, then the rate at which, the flux cuts the conductor is as:

u = d/t

That gives, E = Bld/T

However the total area of the flux that cuts the conductor in the time, T, is the length of conductor, l, in the magnetic field times the distance travelled, d, and hence:

A = ld

And since, BA = Φ

Then, E = BA/T = Φ/T; V (or Wb/s)

On an instantaneous basis, the magnetic flux cutting the conductor could modify itself. This is still, though, the rate at which the flux cuts the conductor that finds out the induced instantaneous emf, e.

In this case: e(t) = dΦ/dt V

Illustration: The faces of poles of a permanent magnet encompass dimensions 150 mm high x 250 mm wide. The 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. Find out the emf induced in a conductor mounted vertically among the faces of the magnet whereas in the magnetic field and the relative polarity of this emf.

Solution:

A = 0.15 x 0.25 m2 = 0.0375 m2

B = Φ/A =0.5 / 0.0375 = 13.3 T

As before:

The length of conductor l that is exposed to the magnetic field in this case is the height of face of the magnet of 150mm.

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

By using the Fleming’s Right Hand Rule, noting that the relative direction of motion of conductor is opposite to that of the magnet, that is, from right to left, it can be established that the emf is induced in conductor acting in an upward direction from bottom to the top.

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