Question

(a) A conductor of length 'I is rotated about one of its ends at a constant
angular speed 'o(omega)' in a plane perpendicular to a uniform magnetic
field B. Plot graphs to show variations of the EMF induced across the
ends of the conductor with (1) angular speed 'o'(omega) and (2) length of the conductor l​

Answer 1

Given:

A conductor of length 'I'.

Angular speed  = ω.

Uniform magnetic  field  = B

To Find:

The EMF induced across the  ends of the conductor and its variation.

Solution:

Let dx be a small element at a distance of x on the rod

The Emf induced in this element due to magnetic field = d∈

Now,

• Velocity of this element = xw

Induced EMF :

• d∈ = B(ωx)dx

Therefore total EMF  = ∫d∈

• ∈ = ∫d∈ = $\int\limits^l_0 {Bwx} \, dx$  = Bωl²/2

Variations of the EMF induced across the  ends of the conductor with

     1. angular speed ω and

If we keep the length of the rod constant,

• ∈ = $\frac{Bl^{2} }{2 }$ .ω
• Therefore ∈ α ω
• Therefore it is a linear relation
• The graph will be a straight line. Fig(a)

     2. length of the conductor l​

If we keep the angular velocity of the rod constant,

• ∈ = $\frac{Bw }{2 }$ .l²
• Therefore ∈ α l²
• Therefore it is a quadratic relation
• The graph will be a Parabola. Fig(b)