Question

A current of 200 $\mu A$ deflects the coil of a moving coil galvanometer through $60^{\circ}$. The current to cause deflection through $\frac{\pi}{10}$ radian is :

• A. 30 $\mu A$
• B. 120 $\mu A$
• C. 60 $\mu A$
• D. 180 $\mu A$

Answer 1

The Correct Option is C

To solve this problem, we need to understand the working principle of a moving coil galvanometer. The deflection in a moving coil galvanometer is directly proportional to the current flowing through it. This is based on the principle that the torque on a coil in a magnetic field is proportional to the current and the number of turns in the coil.

The deflection can be represented as: 

\(\theta \propto I\)

Given:

• A current of 200 \(\mu A\) produces a deflection of \(60^{\circ}\).
• We need to find the current that causes a deflection through \(\frac{\pi}{10}\) radians.

First, convert \(60^{\circ}\) to radians:

\(60^{\circ} = \frac{60 \times \pi}{180} = \frac{\pi}{3} \text{ radians}\)

Using the proportionality, write the equation for both scenarios:

\(\frac{\pi}{3} \propto 200 \mu A\)

\(\frac{\pi}{10} \propto I_{\text{required}}\)

Equating the ratios as the proportionality constant is the same:

\(\frac{200}{I_{\text{required}}} = \frac{\frac{\pi}{3}}{\frac{\pi}{10}}\)

Simplify the right-hand side:

\(\frac{\pi}{3} \cdot \frac{10}{\pi} = \frac{10}{3}\)

Thus, the equation becomes:

\(\frac{200}{I_{\text{required}}} = \frac{10}{3}\)

Cross-multiply to solve for \(I_{\text{required}}\):

\(I_{\text{required}} = \frac{200 \times 3}{10} = 60 \mu A\)

Therefore, the current required to cause a deflection of \(\frac{\pi}{10}\) radians is 60 \(\mu A\).

Hence, the correct option is 60 \(\mu A\).

Answer 2

Given:

- \( i_2 = 200 \, \mu A \),
- \( \theta_2 = 60^\circ = \frac{\pi}{3} \, \text{radians} \).

The deflection \( \theta \) is proportional to the current \( i \). Therefore:

\(\frac{i_1}{i_2} = \frac{\theta_1}{\theta_2}.\)

For \( \theta_1 = \frac{\pi}{10} \, \text{radians} \):

\(\frac{i_1}{200} = \frac{\frac{\pi}{10}}{\frac{\pi}{3}}.\)

Simplify:

\(\frac{i_1}{200} = \frac{3}{10} \implies i_1 = 200 \times \frac{3}{10} = 60 \, \mu A.\)

The Correct answer is: 60 $\mu A$