Chapter Notes: Electromagnetic Induction & Alternating Current Class 12 Physics


Notes for Electromagnetic Induction chapter of class 12 physics. Dronstudy provides free comprehensive chapterwise class 12 physics notes with proper images & diagram.

 

MAGNETIC FLUX

Magnetic flux ({\Phi _B}) through an area d{\rm{\vec S}} in a magnetic field {\rm{\vec B}} is defined as

{\Phi _B} = \int {{\rm{\vec B}}{\rm{.}}d{\rm{\vec S}}}

A magnetic field {\rm{\vec B}} is visualized by drawing lines of field, which are imaginary. However, magnetic flux ({\Phi _B}) is a real physical quantity.
The dimensions of {\Phi _B} can be derived as follows :

[{\Phi _B}] = [B][S] = \left[ {{F \over {AL}}} \right]{\rm{ }}[S]                 [as F = BIL sin\theta ]
or [{\Phi _B}] = \left[ {{{ML{T^{ - 2}}} \over {AL}}} \right][{L^2}] = [M{L^2}{T^{ - 2}}{A^{ - 1}}]

The SI unit of magnetic flux is weber (Wb), which is equivalent to Tm2 or to Vs.
For an elemental area d{\rm{\vec S}} in a magnetic field {\rm{\vec B}}, the associated magnetic flux is given by

d{\Phi _B} = {\rm{\vec B}}{\rm{.}}d{\rm{\vec S = BdScos}}\theta

The flux d{\Phi _B} will be maximum, if \theta = 0, i.e., if the area is perpendicular to the direction of {\rm{\vec B}}. If the area lies along the direction of {\rm{\vec B}} (i.e., d{\rm{\vec S}} is normal to {\rm{\vec B}}), the flux d{\Phi _B} is zero.
Consider a cylinder of radius R placed in a uniform field {\rm{\vec B}}, as shown.

The flux for the right-hand end surface is
{\Phi _B} = + \pi {R^2}B
But for the left-hand end surface, it is
{\Phi _B} = - \pi {R^2}B

Also note that for any area element on the curved surface, {\rm{\hat n}} and {\rm{\vec B}} are mutually perpendicular
(\theta = {90^0}), the magnetic flux linked is {\Phi _B} = 0
As magnetic lines of field are closed curves (i.e., monopoles do not exist), total magnetic flux linked with a closed surface is always zero, i.e.,

\oint {{\rm{\vec B}}} {\rm{.}}d{\rm{\vec S}} = 0

This law is called Gauss’ law for magnetism.

Application 1
At a given place, horizontal and vertical components of earth’s magnetic field BH and BV are along x and y axes respectively as shown. What is the total flux of earth’s magnetic field associated with an area S, if the area S is in (a) x-y plane (b) y-z plane and (c) z-x plane ?
Solution:

Since, here {\rm{\vec B}} = {\rm{\hat i}}{B_H} - {\rm{\hat j}}{B_V} = constant, so

\Phi = \int {{\rm{\vec B}}{\rm{.}}d{\rm{\vec S}} = {\rm{\vec B}}{\rm{.\vec S}}}           [as {\rm{\vec B}} = constt.]

(a) For area S in x-y plane, {\rm{\vec S}} = {\rm{\hat k}}S.

⸫    {\Phi _{xy}} = ({\rm{\hat i}}{B_H} - {\rm{\hat j}}{B_V}){\rm{.}}({\rm{\hat k}}S) = 0          [as {\rm{\hat i}}{\rm{.\hat k}} = {\rm{\hat j}}{\rm{.\hat k}} = 0]

(b) For area S in y-z plane, {\rm{\vec S}} = {\rm{\hat i}}S.

⸫   {\Phi _{yz}} = ({\rm{\hat i}}{B_H} - {\rm{\hat j}}{B_V}){\rm{.}}({\rm{\hat i}}S) = {B_H}S              [as {\rm{\hat i}}{\rm{.\hat i}} = 1 and {\rm{\hat j}}{\rm{.\hat i}} = 0]

(c) For area S in z-x plane, {\rm{\vec S}} = {\rm{\hat j}}S.

⸫   {\Phi _{zx}} = ({\rm{\hat j}}{B_H} - {\rm{\hat j}}{B_V}){\rm{.}}({\rm{\hat j}}S) = - {B_V}S   [as {\rm{\hat i}}{\rm{.\hat j}} = 0 and {\rm{\hat j}}{\rm{.\hat j}} = 1]

INDUCED EMF

FARADAY'S LAWS OF ELECTROMAGNETIC INDUCTION

In 1820, Oersted discovered that an electric current produces magnetic field. This prompted scientists to look for the inverse effect : ‘It must be possible to produce electric current using magnetic field’.

Michael Faraday conducted a series of experiments for eleven long years, and finally on 29th Aug., 1931 he succeeded in producing induced current for the first time. Based on his experiments, he gave following two laws of electromagnetic induction (EMI) :

Law I :

Whenever there is a change of flux linked with a circuit, or whenever a moving conductor cuts the flux, an emf is induced in it.
This phenomenon is called electromagnetic induction and the emf, induced emf. If the circuit is closed the current which flows in it due to induced emf is called induced current.

Law II :

The magnitude of induced emf is equal to the rate of change of flux, i.e.,

E = \left| {{{d\Phi } \over {dt}}} \right|{\rm{ }}

The direction or the sense of the induced emf (or induced current) is given by Lenz’s law.            

Eddy current

When a metallic conducting sheet is moved in magnetic field, A current loop is developed on its surface which is called eddys current. Due to eddy’s current, thermal energy is produced in it. This energy is reduced at the loss of kinetic energy of the plate and the plate slows down. It is known as electromagnetic damping.

LENZ'S LAW

The effect of the induced emf is such as to oppose the change in flux that produces it.
In other words, if the flux decreases, then the induced current tries to support the existing magnetic field. If the flux increases, the induced emf tries to decrease the existing magnetic field by creating field in opposite direction.
Considering all the above statements, the Faraday’s laws of electromagnetic induction can now be analytically expressed as

E = - {{d\Phi } \over {dt}}

Negative sign indicates towards the Lenz’s law.
Note that in case of electromagnetic induction, an emf always exists whether the circuit is closed or open. But, the induced current will exist only if the circuit is closed. If the total resistance of the circuit is R, the induced current is

i = {E \over R}

Induced Charge Flow

When a current is induced in the circuit due to the flux change, charge flows through the circuit. The net amount of charge which flows along the circuit is given as :

i = {E \over R} = {1 \over R}{{d\Phi } \over {dt}}         or        {{dq} \over {dt}} = {1 \over R}{{d\Phi } \over {dt}}       or        dq = {1 \over R}d\Phi
⸫    q = \int_{{\rm{ }}0}^{{\rm{ }}q} {} dq = \int_{{\rm{ }}{\Phi _1}}^{{\rm{ }}{\Phi _2}} {} {1 \over R}d\Phi = {{{\Phi _2} - {\Phi _1}} \over R} = {{\Delta \Phi } \over R}

Thus, the induced charge is independent of the manner and time in which the flux changes. However, the induced emf and current depend on time.

FARADAY'S EXPERIMENTS

The experiments performed by Faraday can be categorized into following three groups.

Group I :
Two coils are coaxially arranged such that the magnetic flux produced by one links with the other. The first coil (called the primary coil) is connected to a battery, a switch and a rheostat. The second coil (called the secondary coil) is connected to galvanometer.When switch S is closed and the current in the primary coil is increased, there is an induced emf and current in the secondary [Fig. (A)]. Again when the current in the primary is decreased by increasing the resistance R of the rheostat, the galvanometer G shows deflection in the opposite direction [Fig. (B)]. This time the emf and the current is induced in the secondary in the same direction as that in the primary.
Surprising thing is that when the current in primary remains steady, there is no emf or current induced in the secondary.

When the current in the primary increases [Figure (A)], the magnetic flux produced by it also increases. Now, as per Lenz’s law, the induced current in the secondary should be such as to reduce the field B. The direction of this induced current is shown in figure (A).
In figure (B), the current in primary decreases thereby decreasing field B. Therefore, as per Lenz’s law, induced current in the secondary will be in a direction so as to increase the field B. The direction of this induced current is shown in figure (B).

Group II :
A coil is arranged to link some of the magnetic flux from a source (which may be either a magnet or a current).If the source is moved towards the coil, the flux or field B increases. The induced emf or the current in the coil will be in such a direction so as to try to reduce the field [figure (A)].
If the source is moved away [figure (B)], the field B decreases, and as a result a current as shown in figure (B) is induced in the coil in such a direction so as to try to increase the field.

Note that the magnitude and the direction of the induced current depend on the relative velocity of the source and the coil. It is immaterial whether the source moves, or the coil moves or both of them move. The important thing for the emf to be induced is that there should be change of flux linked with the coil.

Group III :
If part of a conducing circuit is moving and there by cutting magnetic flux, a current is induced in the circuit. In figure (A), rod AB slides over the rails, and moves toward right with a velocity, v, thereby cutting the magnetic flux. Induced current flows from B to A.In figure (B),  a rod AB rotates with angular velocity \omega  so as to cut the magnetic flux. Again, induced current flows from B to A. In figure (C), a disc is made to rotate in a magnetic field. Current flows in the circuit because of induced emf. In fact, this is known as Faraday’s disc. It was the first continuous generator given to the world by Faraday.



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