Two moving coil galvanometers `1` and `2` are with identical field magnets and suspension torque constants, but with coil of different number of turns `N_(1)` and `N_(2)`, area per turn `A_(1)` and `A_(2)`, and resistance `R_(1)` and `R_(2)`. When they are connected in series in the same circuit, they show deflections `theta_(1)` and `theta_(2)`. then `theta_(1)//theta_(2)` is

Options

A. `A_(1) N_(1)// A_(2) N_(2)`
B. `A_(1) N_(2)// A_(2) N_(1)`
C. `A_(1) R_(2) N_(1)// A_(2) R_(2) N_(2)`
D. `A_(1)R_(1) N_(1)// A_(2)R_(2) N_(2)`

Explanation

To solve the problem, we will analyze the relationship between the deflections of the two galvanometers when they are connected in series.

### Step-by-Step Solution:

1. **Understanding the Setup**:
Two moving coil galvanometers (G1 and G2) are connected in series in the same circuit. They have identical field magnets and suspension torque constants, but differ in the number of turns (N1, N2), area per turn (A1, A2), and resistance (R1, R2).

2. **Current in Series Connection**:
Since the galvanometers are connected in series, the current (I) flowing through both galvanometers is the same. Thus, we can write:
\[ I_1 = I_2 \]

3. **Current Expression for Galvanometers**:
The current through a galvanometer can be expressed as:
\[ I = \frac{k \theta}{N \cdot B \cdot A} \]
where \( k \) is a constant, \( \theta \) is the deflection, \( N \) is the number of turns, \( B \) is the magnetic field strength, and \( A \) is the area per turn.

4. **Writing the Current Equations**:
For galvanometer 1:
\[ I_1 = \frac{k \theta_1}{N_1 \cdot B \cdot A_1} \]
For galvanometer 2:
\[ I_2 = \frac{k \theta_2}{N_2 \cdot B \cdot A_2} \]

5. **Equating the Currents**:
Since \( I_1 = I_2 \), we can set the two equations equal to each other:
\[ \frac{k \theta_1}{N_1 \cdot B \cdot A_1} = \frac{k \theta_2}{N_2 \cdot B \cdot A_2} \]

6. **Canceling Common Terms**:
The constants \( k \) and \( B \) appear on both sides of the equation, so they can be canceled out:
\[ \frac{\theta_1}{N_1 \cdot A_1} = \frac{\theta_2}{N_2 \cdot A_2} \]

7. **Rearranging the Equation**:
Rearranging gives us the relationship between the deflections:
\[ \frac{\theta_1}{\theta_2} = \frac{N_1 \cdot A_1}{N_2 \cdot A_2} \]

8. **Final Result**:
Therefore, the final expression for the ratio of the deflections is:
\[ \frac{\theta_1}{\theta_2} = \frac{N_1 A_1}{N_2 A_2} \]

### Conclusion:
The ratio of the deflections of the two galvanometers when connected in series is given by:
\[ \frac{\theta_1}{\theta_2} = \frac{N_1 A_1}{N_2 A_2} \]