A tangent galvanometer has a coil with 50 turns and radius equal to 4 cm . A current of 0.1 A is passing through it. The plane of the coil is set parallel to the earth's magnetic meridian. If the value of the earth's horizontal component of the magnetic field is `7xx10^(-5)` Tesla and `mu_(0)=4pixx10^(-7)Weber//ampxxm` then the deflection in the galvanometer needle will be

Options

A. `45^(@)`
B. `48.2^(@)`
C. `50.7^(@)`
D. `52.7^(@)`

Explanation

To solve the problem, we need to find the deflection angle (θ) of the galvanometer needle using the given parameters. We can use the formula for the tangent galvanometer:

\[ \tan \theta = \frac{\mu_0 n I}{2 R B_h} \]

Where:
- \( \mu_0 = 4 \pi \times 10^{-7} \, \text{Weber/Ampere meter} \)
- \( n = 50 \) (number of turns)
- \( I = 0.1 \, \text{A} \) (current)
- \( R = 4 \, \text{cm} = 0.04 \, \text{m} \) (radius)
- \( B_h = 7 \times 10^{-5} \, \text{T} \) (horizontal component of the Earth's magnetic field)

### Step 1: Substitute the values into the formula

First, we need to substitute the known values into the formula:

\[ \tan \theta = \frac{(4 \pi \times 10^{-7}) \times 50 \times 0.1}{2 \times 0.04 \times (7 \times 10^{-5})} \]

### Step 2: Calculate the numerator

Calculating the numerator:

\[ 4 \pi \times 10^{-7} \times 50 \times 0.1 = 2 \pi \times 10^{-7} \times 5 = 10 \pi \times 10^{-7} \]

Using \( \pi \approx 3.14 \):

\[ 10 \pi \times 10^{-7} \approx 31.4 \times 10^{-7} = 3.14 \times 10^{-6} \]

### Step 3: Calculate the denominator

Now, calculating the denominator:

\[ 2 \times 0.04 \times (7 \times 10^{-5}) = 0.08 \times 7 \times 10^{-5} = 5.6 \times 10^{-6} \]

### Step 4: Calculate \( \tan \theta \)

Now substituting the values back into the equation:

\[ \tan \theta = \frac{3.14 \times 10^{-6}}{5.6 \times 10^{-6}} \approx 0.5607 \]

### Step 5: Find \( \theta \)

To find \( \theta \), we take the arctangent:

\[ \theta = \tan^{-1}(0.5607) \]

Using a calculator, we find:

\[ \theta \approx 29.5^\circ \]

### Conclusion

The deflection in the galvanometer needle will be approximately \( 29.5^\circ \).

---