Electromagnetic Induction

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Writing the expression for magnetic flux:

Laws of Electromagnetic induction are heavily centered around the concept of magnetic flux

ϕ_{B}

and its variation with time. So it is very important to learn how to write magnetic flux through any closed area. The area is considered as vector.
We can look at the expression B A

cos θ

in two ways:

B (A $cos θ$ ) i.e., B times the projection of area normal to B, or

B $⊥$ A, i.e., component of B along the normal to the area element times the magnitude of the area element.

If the magnetic field has different magnitudes and directions at various parts of a surface as shown in, then the magnetic flux through the surface is given by

Φ_{B} = B_{1} . d A_{1} + B_{2} . d A_{2} + . . . = \int B . d A

Let us have a better understanding of magnetic flux crossing an area A through this video.

Introduction of Magetic flux

7 mins

Energy stored in inductor:
Whenever a current is established in the circuit, the inductance opposes the growth of the current. In order to establish a current in the circuit, work is done against this opposition by some external agency. This work done is stored as magnetic potential energy. This stored energy is the reason why self inductance plays the role of inertia and opposes any change (increase or decrease) of current in the coil.

Let us derive the expression for energy stored in an inductor through this video.

Energy stored in inductor

6 mins

Motional EMF:
EMF can be induced by changing:

the magnetic field

the area of the coil

the relative orientation of coil w.r.t. magnetic field

We can induce emf by moving a conductor instead of varying the magnetic field, that is, by changing the magnetic flux enclosed by the circuit. The induced emf is called motional emf.
It can be explained by Faraday's Law as the area linked with the magnetic flux is changing.

It is also possible to explain the motional emf by invoking the Lorentz force acting on the free charge carriers of conductor. When the rod moves with speed v, the charge will also be moving with speed v in the magnetic field B. The Lorentz force on this charge is qvB in magnitude, and its direction is downwards. All charges experience the same force, in magnitude and direction, irrespective of their position in the rod PQ. The work done in moving the charge from P to Q is,

W = q v B l

Since emf is the work done per unit charge,

ε = \frac{W}{q} = B l v

The video will explain both the approaches to derive the expression for motional emf.

Class 12 Physics | #6 EMF Induced in a Moving Conductor in Uniform Magnetic Field-Motional EMF

8 mins

The axis of rotation of the coil is perpendicular to the direction of the magnetic field. The coil (called armature) is mechanically rotated in the uniform magnetic field by some external means. The rotation of the coil causes the magnetic flux through it to change, so an emf is induced in the coil.

This video will explain the derivation of induced EMF equation for an AC generator.

AC Generator

2 mins