Question

A 20 cm long metallic rod is rotated with 210 rpm about an axis normal to the rod passing through its one end. The order end of the rod is in contact with a circular metallic ring. A constant and uniform magnetic field 0.2T parallel to the axis exists everywhere. The emf developed between the centre and the ring is ____ mV.

Answer 1

Correct Answer: 88

We are given the length of a rod and its angular speed. We need to find the induced emf across the ends of the rod in a magnetic field.

Solution

1. Given Values:

Length of the rod, \( \ell = 20 \, \text{cm} = 0.2 \, \text{m} \)

Angular speed, \( \omega = 210 \, \text{rpm} \)

Magnetic field, \( B = 0.2 \, \text{T} \) (assumed from the calculation)

2. Convert rpm to Radians per Second:

\( \omega = 210 \times \frac{2\pi}{60} = 210 \times \frac{\pi}{30} = 7\pi \, \text{rad/s} \)

Using the approximate value of π as 22/7:

\( \omega = 7 \times \frac{22}{7} = 22 \, \text{rad/s} \)

3. Use the Formula for Induced EMF:

The induced emf is given by:

\( \text{emf} = \frac{1}{2} B \omega \ell^2 \)

4. Substitute the Known Values:

\( \text{emf} = \frac{1}{2} \times 0.2 \times 22 \times (0.2)^2 \)

\( \text{emf} = 0.1 \times 22 \times 0.04 \)

\( \text{emf} = 2.2 \times 0.04 \)

\( \text{emf} = 0.088 \, \text{V} \)

\( \text{emf} = 88 \, \text{mV} \)

Final Answer

Thus, the induced emf is 88 mV.