Two moving coil galvanometers A and B have rectangular coil of 100 turns each. The area of coil of galvanometers A and B are `2 xx 10^(-4) m^(2) " and " 6 xx 10^(-4) m^(2)` respectively. Calculate the ratio of current sensitivity of A to B if coils in both salvanometers are kept in a radial horizontal magnetic field of `0.2` T and the torsional constant of hair springs are also same.

Answer

To find the ratio of current sensitivity of galvanometers A and B, we can follow these steps:

### Step 1: Understand the formula for current sensitivity
The current sensitivity (Is) of a moving coil galvanometer is given by the formula:
\[ Is = \frac{NAB}{K} \]
where:
- \(N\) = number of turns in the coil
- \(A\) = area of the coil
- \(B\) = magnetic field strength
- \(K\) = torsional constant of the spring

### Step 2: Identify the given values
For both galvanometers A and B:
- Number of turns, \(N = 100\)
- Area of coil for A, \(A_1 = 2 \times 10^{-4} \, m^2\)
- Area of coil for B, \(A_2 = 6 \times 10^{-4} \, m^2\)
- Magnetic field strength, \(B = 0.2 \, T\)
- Torsional constant, \(K\) is the same for both galvanometers.

### Step 3: Write the expressions for current sensitivity
For galvanometer A:
\[ Is_A = \frac{N \cdot A_1 \cdot B}{K} \]
For galvanometer B:
\[ Is_B = \frac{N \cdot A_2 \cdot B}{K} \]

### Step 4: Calculate the ratio of current sensitivity
To find the ratio of current sensitivity \( \frac{Is_A}{Is_B} \):
\[ \frac{Is_A}{Is_B} = \frac{\frac{N \cdot A_1 \cdot B}{K}}{\frac{N \cdot A_2 \cdot B}{K}} = \frac{A_1}{A_2} \]
Since \(N\), \(B\), and \(K\) are the same for both, they cancel out.

### Step 5: Substitute the values of areas
Now substitute the values of \(A_1\) and \(A_2\):
\[ \frac{Is_A}{Is_B} = \frac{2 \times 10^{-4}}{6 \times 10^{-4}} = \frac{2}{6} = \frac{1}{3} \]

### Final Result
Thus, the ratio of current sensitivity of A to B is:
\[ \frac{Is_A}{Is_B} = 1 : 3 \]