Question

A conductor of length 'l is rotated about one of its ends at a constant
angular speed 'o' 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 (i) angular speed a and (ii) length of the
conductor l.​

Answer 1

Given:

Length of the conductor =  I m.

Angular speed of rotation   = a rad/s.

Uniform magnetic  field  = B Tesla.

To Find:

The EMF induced across the  ends of the conductor and its variation with angular speed and the length of the conductor.

Solution:

Let dr be a small element, which is at a distance  r meters from the pivot point of the rod

Let dE be the Emf induced in this element due to the uniform magnetic field B.

Now,

Velocity of this small  element dr  = ra

Then the Induced EMF :

• d∈ = B(ra)dr

Therefore total EMF,

• E  = ∫dE

• E = ∫dE = $\int\limits^l_0 {aBr} \, dr$  = Bal²/2

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

             1. angular speed ω and

If we assume the length of the rod to be constant,

• E =  (Bl²/2).a

Therefore

• E α a

• Hence it is a linear relation

• The graph will be a straight line. Fig(1)

    2. length of the conductor l​

If we assume the angular speed of the rod to be constant,

E =  (Ba/2) l²

• Therefore E α l²

• Hence it is a quadratic relation

• The graph will be a Parabola. Fig(2)

The variation of emf induced across the ends of the rod with angular speed is linear.

The variation of the emf induced across the ends of the rod with the length of the rod is quadratic.