Two tangent galvanometers have coils of the same radii, differing only in their number of turns. They are connected in series. When a steady current is passed in the circuit, the mean deflections in the galvanometer are `theta_1` and `theta_2`. Deduce an expression for the ratio of number of turns of the galvanometer.

Explanation

To solve the problem, we need to derive an expression for the ratio of the number of turns of the two tangent galvanometers based on the given deflections when a steady current is passed through them.

### Step-by-Step Solution:

1. **Understanding the Tangent Galvanometer**:
A tangent galvanometer consists of a coil of wire that produces a magnetic field when current flows through it. The deflection angle (θ) of the needle is related to the current (I) and the number of turns (n) in the coil.

2. **Formula for Tangent Galvanometer**:
The current in a tangent galvanometer can be expressed as:
\[ I = \frac{2rH}{\mu_0 n \tan \theta} \]
where:
- \( r \) is the radius of the coil,
- \( H \) is the horizontal component of the Earth's magnetic field,
- \( \mu_0 \) is the permeability of free space,
- \( n \) is the number of turns,
- \( \theta \) is the deflection angle.

3. **Setting Up the Equations**:
For the first galvanometer with deflection \( \theta_1 \) and number of turns \( n_1 \):
\[ I_1 = \frac{2rH}{\mu_0 n_1 \tan \theta_1} \]
For the second galvanometer with deflection \( \theta_2 \) and number of turns \( n_2 \):
\[ I_2 = \frac{2rH}{\mu_0 n_2 \tan \theta_2} \]

4. **Connecting the Galvanometers in Series**:
Since the galvanometers are connected in series, the current flowing through both is the same:
\[ I_1 = I_2 \]

5. **Equating the Two Currents**:
Setting the two expressions for current equal gives:
\[ \frac{2rH}{\mu_0 n_1 \tan \theta_1} = \frac{2rH}{\mu_0 n_2 \tan \theta_2} \]

6. **Cancelling Common Terms**:
We can cancel \( 2rH \) and \( \mu_0 \) from both sides:
\[ \frac{1}{n_1 \tan \theta_1} = \frac{1}{n_2 \tan \theta_2} \]

7. **Rearranging the Equation**:
Rearranging gives us:
\[ n_1 \tan \theta_1 = n_2 \tan \theta_2 \]

8. **Finding the Ratio of Number of Turns**:
Dividing both sides by \( n_2 \tan \theta_2 \) results in:
\[ \frac{n_1}{n_2} = \frac{\tan \theta_2}{\tan \theta_1} \]

### Final Expression:
The ratio of the number of turns of the two galvanometers is given by:
\[ \frac{n_1}{n_2} = \frac{\tan \theta_2}{\tan \theta_1} \]