Question

A moving coil galvanometer has 100 turns and each turn has an area of $2.0 \, \text{cm}^2$. The magnetic field produced by the magnet is $0.01 \, \text{T}$ and the deflection in the coil is $0.05$ radians when a current of $10 \, \text{mA}$ is passed through it. The torsional constant of the suspension wire is $x \times 10^{-5} \, \text{N} \cdot \text{m/rad}$. The value of $x$ is:

Answer 1

Given: - Number of turns: $N = 100$ - Area of each turn: $A = 2.0 \, \text{cm}^2 = 2.0 \times 10^{-4} \, \text{m}^2$ - Magnetic field: $B = 0.01 \, \text{T}$ - Deflection: $\theta = 0.05 \, \text{radian}$ - Current: $I = 10 \, \text{mA} = 10 \times 10^{-3} \, \text{A}$

Step 1: Calculating the Torque Acting on the Coil

The torque ($\tau$) acting on a moving coil galvanometer is given by:

$\tau = N \times B \times I \times A$

Substituting the given values:

$\tau = 100 \times 0.01 \, \text{T} \times 10 \times 10^{-3} \, \text{A} \times 2.0 \times 10^{-4} \, \text{m}^2$ $\tau = 100 \times 0.01 \times 0.01 \times 2.0 \times 10^{-4} \, \text{N} \times \text{m}$ $\tau = 2 \times 10^{-5} \, \text{N} \times \text{m}$

Step 2: Relating Torque to the Torsional Constant

The torque is also related to the torsional constant ($k$) and deflection ($\theta$) by:

$\tau = k \times \theta$

Rearranging to find $k$:

$k = \frac{\tau}{\theta}$

Substituting the given values:

$k = \frac{2 \times 10^{-5}}{0.05} \, \text{N} \times \text{m/rad}$ $k = 4 \times 10^{-4} \, \text{N} \times \text{m/rad}$

Step 3: Finding the Value of $x$

Given that $k = x \times 10^{-5} \, \text{N} \times \text{m/rad}$:

$4 \times 10^{-4} = x \times 10^{-5}$

Solving for $x$:

$x = 4$

Conclusion: The value of $x$ is 4.