Question

(i) What is induced emf? Write Faraday’s law of electromagnetic induction. Express it mathematically. 

(ii) A conducting rod of length ‘l’, with one end pivoted, is rotated with a uniform angular speed ‘ω’ in a vertical plane, normal to a uniform magnetic field ‘B’. Deduce an expression for the emf induced in this rod.

If resistance of rod is R, what is the current induced in it?

Answer 1

(i) Induced emf: The emf developed in a coil due to change in magnetic flux linked with the coil is called the induced emf. 

Faraday’s Law of Electromagnetic Induction: On the basis of experiments, Faraday gave two laws of electromagnetic induction: 

(i) When the magnetic flux linked with a coil or circuit changes, an emf is induced in the coil. If coil is closed, the current is also induced. The emf and current last so long as the change in magnetic flux lasts. The magnitude of induced emf is proportional to the rate of change of magnetic flux linked with the circuit. Thus if ∆φ is the change in magnetic flux linked in time ∆t then rate of change of flux is

\(\frac{∆φ}{∆t}\),

So emf induced ε ∝ \(\frac{∆φ}{∆t}\)

2. The emf include in the coil (or circuit) opposes the cause producing it.

 ε ∝ -\(\frac{∆φ}{∆t}\)

Here the negative sign shows that the included emf ε opposes the change in magnetic flux.

 ε =-K \(\frac{∆φ}{∆t}\) where K is a constant of proportionality which depends on units chosen for φ,t and ε. In SI system the unit of flux φ is weber, unit of time t is second and unit of emf ε' is volt and K = I

ε = - \(\frac{∆φ}{∆t}\) .....(i)

If the coil contains N- turns of insulated wire, then the flux linked with each turn will be same and the emf include in each turn will be in the  same direction,hence the emfs of all turns will be added. Therefore the emf induced in the whole coil.

ε = - N\(\frac{∆φ}{∆t}\) = - \(\frac{∆(Nφ)}{∆t}\)  ....(ii)

Nφ is called the effective magnetic flux or the number of flux linkages in the coil and may be denoted by φ 

(ii) Expression for Induced emf in a Rotating Rod

Consider a metallic rod OA of length l which is rotating with angular velocity ω in a uniform magnetic field B, the plane of rotation being perpendicular to the magnetic field. A rod may be supposed to be formed of a large number of small elements. Consider a small element of length dx at a distance x from centre. If v is the linear velocity of this element, then area swept by the element per second = v dx 

The emf induced across the ends of element