Two tangent galvanometers having coils of the same radius are connected in series. A current flowing in them produces deflections of `60^(@)` and `45^(@)` respectively. The ratio of the number of turns in the coils is

Options

A. `4//3`
B. `(sqrt(3)+1)//1`
C. `sqrt(3)//1`
D. `(sqrt(3)+1)//(sqrt(3)-1)`

Explanation

To solve the problem step by step, we will use the principles of tangent galvanometers and the tangent law of magnetism.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two tangent galvanometers connected in series, producing deflections of \(60^\circ\) and \(45^\circ\). We need to find the ratio of the number of turns in their coils.

2. **Using the Tangent Law**:
According to the tangent law, the magnetic field \(B\) produced by a tangent galvanometer is given by:
\[ B = \mu_0 \frac{N I}{2a} \]
where \(N\) is the number of turns, \(I\) is the current, and \(a\) is the radius of the coil.

3. **Setting Up the Equations**:
For the first galvanometer (deflection \(60^\circ\)):
\[ B_1 = \mu_0 \frac{N_1 I}{2a} = B_h \tan(60^\circ) \]
For the second galvanometer (deflection \(45^\circ\)):
\[ B_2 = \mu_0 \frac{N_2 I}{2a} = B_h \tan(45^\circ) \]

4. **Finding the Ratio of Magnetic Fields**:
From the above equations, we can express the ratio of the magnetic fields:
\[ \frac{B_1}{B_2} = \frac{\tan(60^\circ)}{\tan(45^\circ)} \]

5. **Calculating the Tangents**:
We know:
\[ \tan(60^\circ) = \sqrt{3}, \quad \tan(45^\circ) = 1 \]
Therefore:
\[ \frac{B_1}{B_2} = \frac{\sqrt{3}}{1} = \sqrt{3} \]

6. **Relating the Number of Turns**:
Since the magnetic fields are proportional to the number of turns:
\[ \frac{N_1}{N_2} = \frac{B_1}{B_2} = \sqrt{3} \]

7. **Final Ratio**:
Thus, the ratio of the number of turns in the coils is:
\[ \frac{N_1}{N_2} = \sqrt{3} : 1 \]

### Conclusion:
The ratio of the number of turns in the coils of the two galvanometers is \(\sqrt{3} : 1\).