Electro Magnetic Induction

ElectroMagnetic Induction:

A charged body is capable of producing electric charge in a neighbouring conductor. The phenomenon of induction of electricity due to electricity is called electric induction. A magnet is capable of producing magnetism in a neighbouring magnetic substance. This phenomenon is production of magnetism due to magnetism is called magnetic induction. A current flowing through a wire produces a magnetic field around itself. This phenomenon of production of magnetism due to electricity is called magnetic effect of currents. The production of electricity due to magnetism is called electro-magnetic induction. 

 Magnetic Flux:

'Flux' is a word used in a study of the quantity of certain fluids following across any area. Magnetic flux deals with the study of the number of lines of force of magnetic field crossing a certain area. 

Consider an area 'a' placed in a magnetic field having magnetic induction 'B'. Let the area be inclined to the direction of 'B' at an angle 'θ' (Fig. 1). 

Fig. 1. Lines of force of magnetic field cutting through an area. 

The area, in vector notation, can be represented by a vector directed along the normal to the area and having a length proportional to the magnitude of the area. Magnetic flux 'ΦB' through area 'a' is given by

    ΦB  = B. A = BA cos θ = A (B cos θ) 

But, B cos θ = component of B perpendicular to the area 'a'. 

Magnetic flux linked with a surface, is defined as the product of area and the component of 'B' perpendicular to the area. 

Case 1. If θ = 90°, θ = 0

When angle between 'B' and the normal to the surface is 90° 'B' will be parallel to the surface (Fig. 2).

Fig. 2. Surface held parallel to lines of force. 

∴     ΦB   = BA × 0 = 0.

No magnetic flux is linked with the surface when the field is parallel to the surface. 

Case 2. If θ = 0° , cos θ = 1.

In this case 'B' is perpendicular to the surface (Fig. 3).

        (ΦB) = B × A × 1 = BA. 

Magnetic flux linked with a surface is maximum when area is held perpendicular to the direction of field. 

     Since      B = μH

Where 'μ' is the permeability of the medium and H is magnetic intensity, i.e., the number of lines of force per unit area in free space. 

     (dΦB)max = μAH

In case the magnetic field is not uniform or the surface is not a plane one, the magnetic flux ΦB linked with a surface can be obtained as follows. 

Let '(dΦB)' be the magnetic flux linked with a small area dA of the surface. The area is so small that the field can be considered to be the same over that area. 

            dΦB = B. dA = B. n̂dA

where n̂ is a unit vector alonɡ the outward drawn normal to the surface at that point.

∴      ΦB = B0∫ dΦB = S∫ B. dA

or ΦB = S∫ B. n̂dA

Thus, magnetic flux linked with a surface in a magnetic field is defined as the surface integral of the magnetic flux density over that surface. 

Any line of force cutting an electric circuit more than once has its effect multiplied by the number of times it cuts the circuit. 

One line cutting an electric circuit five times [Fig. 4(i)] is equivalent to 5 lines cutting the electric circuit once [Fig. 4(ii)]. In both the circuits magnetic flux is same. 

Fig. 4. Magnetic flux through a coil. 

If a line of force cuts the circuit n-times, i.e., it passes through a coil having n turns, total magnetic flux linked with it is, 

              ΦB = BA × n

or ΦB = μ nAH

Units of Magnetic flux

(i) S.I. unit. 

When an area 'a' is held perpendicular to the direction of lines of force, magnetic flux linked with it is

              ΦB = BA

If      B = 1 tesla, A = 1 m², ΦB = 1 × 1 weber. 

Weber. Magnetic flux linked with an area of 1 m² held normal to the direction of lines of force of a magnetic field of strength 1tesla is called 1 weber. 

(ii) C.G.S. unit

If  B = 1 gauss,        A = 1 cm²

                                 ΦB = 1 × 1 = 1 maxwell. 

Maxwell. Magnetic flux linked with an area of 1 cm² held normal to the direction of lines of force of a magnetic field of strength 1 guess is called 1 maxwell. 

           1 maxwell = 1 gauss cm²

Relation between weber and maxwell

         1 weber = 1 tesla × 1 m²

                        = 10⁴ gauss × (100 cm)²

                        = 10⁸ gauss cm²

∴      1 weber = 10⁸ maxwell

Dimensional formula of  ΦB :

                 ΦB = B . A = BA cos θ

or             ΦB = F/qv  A cos θ

where F is the force experienced by a change q moving with velocity v in the magnetic field. 

∴           [ΦB] = [M1 L1 T-2] × [L2]/[A1T1] [L T-1]

or [ΦB] = [M1 L2 T-2 A-1]

which is the required formula. 

The dimensions of magnetic flux are 1, 2, -2 and -1 in mass, length, time and electric current respectively. 

Important notes 1. Magnetic flux may or may not be equal to the number of magnetic lines of force passing through the circuit.  2. Magnetic flux ΦB linked with a surface depends upon the angle 'θ' between B and the normal to the surface.  (a) If θ = 0°, ΦB is maximum.  (b) If θ = 90°, ΦB is zero.  3. Magnetic flux linked with a coil, placed in a magnetic field, depends upon the permeability of the material placed inside the coil.

Faraday's Laws of Electromagnetic Induction:

 Faraday's law deal with the induction of an e.m.f. in an electric circuit when magnetic flux linked with the circuit changes. They are started as follows :

Faraday's first law (qualitative) :

Whenever magnetic flux linked with a circuit changes, an e.m.f. is induced in it. 

The induced e.m.f. exists in the circuit so long as the change in magnetic flux linked with it continues. 

Faraday's second law (quantitative) :

The induced e.mf. is directly to the negative rate of change in magnetic flux linked with the circuit. 

If 'dΦB' is the change in magnetic flux linked with a circuit, that takes place in a time dt.  

       Rate of change of magnetic flux = dΦB/dt.

If 'E' is e.m.f. induced in the circuit as a result of this change, 

E ∝ -dΦB /dt       or      E = - K  dΦB/dt

By selecting unit of 'E', 'ΦB' and 't' in a proper way, we can have

   K = 1           ∴         E = - dΦB/dt

     Negative sign is due to direction of induced e.m.f., which is explained by Lenz's law. 

Experiment verification:

     Consider a coil 'L' wound over a hollow wooden cylinder and having 'n' number of turns. It has a galvanometer 'G' connected between its free terminals. A magnet  N-S is placed in its neighbourhood with its center at 'A' [in Fig. 1]. The coil has some magnetic flux linked with it when magnet is at 'A'. Now move the magnet towards the coil so that its center moves from 'A' to 'B'. As a result of this, higher intensity region of magnetic field has moved closer to the coil, thereby, increasing 'H' and hence increasing magnetic flux 'ΦB' (= μnaH) linked with the circuit.

Fig. 1. Change in magnetic flux, linked with a coil, due to motion of a magnet. 

     First law. It will be observed that as magnet is moved from 'A' to 'B', galvanometer gives a deflection. This indicates that electric current and e.m.f. has been induced in the circuit due to a change in magnet flux linked with it. This verifies the first law. It will be observed that the deflection in the galvanometer persists so long as the magnet is in motion from 'A' to 'B' or from 'B' to 'A'. As soon as the magnet comes to rest, deflection disappears. This verifies the second law. 

     Second law. Bring the magnet from A to B at such a speed that it takes 2 seconds to reach B. Note the deflection in galvanometer. Now take the magnet from B to A in 1 second. It will be observed that the deflection obtained in second case is double of that obtained in first case. Since rate of change of magnetic flux in second case is double than that in first case. Hence, e.m.f. is directly proportional to the rate of change of magnetic flux. This varifies the third law. 

     Magnetic flux linked with a circuit can be changed in a number of ways :

• By moving a magnet towards a fixed coil or moving it away from coil. 
• By moving a coil towards a fixed magnet or moving it away from magnet. 
• By increasing or decreasing current in a neighborhood circuit. 

Faraday's Experiments:

It observed by Faraday's that whenever magnetic flux linked with a circuit changes, an e.m.f. is induced in the circuit. If the circuit is closed, it will cause an electric current to flow through the circuit. There may be different reasons for the changing magnetic flux, in different experiments, but the result is same. 

     (i) Induction due to motion of magnet. Consider a coil comprising of a large number of turns of a wire wound over an insulating cylinder. A galvanometer is connected in between its free terminals (Fig. 2). 

Fig. 2. Motion of magnet near a coil. 

     Let a magnet having its North Pole pointing towards the cross-section of the coil be moved towards it [Fig. 2(i)]. As the magnet approaches the coil the strength of magnetic field around the coil increases. This results in an increase in magnetic flux. It will be observed that the galvanometer connected in the circuit gives a deflection indicating the induction of electric current. It can further be observed that the direction of deflection, in galvanometer, gets reversed if the North pole of the magnet is moved away from the face of the coil [Fig. 2(ii)]. Thus, electric current is induced due to motion of magnet. 

     (ii) Induction due to motion of coil. The phenomenon, explained above, can also be observed if the coil having a galvanometer connected in its circuit is moved in the magnetic field of a stationary magnet (Fig. 3). It can also be noted that the direction of deflection in the galvanometer gets reversed if we reverse the direction of motion of coil. In this case also, the induction of electric current takes place due to a change in magnetic flux which takes place due to motion of coil. 

Fig. 3. Motion of coil near a magnet. 

     (iii) Induction due to a current changing in the neighborhood. Consider two coils P and S would over each other with an insulating cylindrical core. The two coils are electrically insulated from each other. A source of d.c. is connected in the circuit of coli P while a galvanometer is connected in the circuit of coil S (Fig. 4).

Fig. 4. A current changing in a neighbouring circuit. 

      It will be observed that the galvanometer gives a deflection when the circuit of coil P is made or broken. The deflection at these two instances are in opposite directions and these are inspite of the fact that there is no electrical contact between P and S. This is gain due to the change in magnetic flux linked with coil S. This time the change in magnetic flux takes place due to a change in strength of magnetic field, around S, due to a changing current in P which changes from zero to maximum at make and from maximum to zero at break. If the key K is kept pressed, a constant current will flow through P. There will be no change in magnetic flux linked with S. Therefore, there is no deflection in the galvanometer.