To know the resistance G of a galvanometer by half deflection method,a battery of emf `V_(E)` and resistance R is used to deflect the galvanometer by angle `theta`.If a shunt of resistance S in needed get half deflection then G,R and S arc related by the equation

Options

A. `S(R+G)=RG`
B. `2S(R+G)=RG`
C. `2G=S`
D. `2S=G`

Answer

To find the relationship between the resistance \( G \) of a galvanometer, the resistance \( R \), and the shunt resistance \( S \) using the half deflection method, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- A galvanometer with resistance \( G \) is connected in parallel with a shunt resistance \( S \).
- A battery of emf \( V_E \) and a resistance \( R \) are connected in series to the galvanometer.

2. **Current Before Adding Shunt**:
- The total current \( I_1 \) flowing through the circuit before the shunt is added can be expressed as:
\[ I_1 = \frac{V_E}{R} \]

3. **Current After Adding Shunt**:
- When the shunt \( S \) is added, the current through the galvanometer becomes half, i.e., \( \frac{I_1}{2} \).
- The current through the shunt \( S \) will then be \( I - \frac{I_1}{2} \).

4. **Voltage Across Galvanometer and Shunt**:
- The voltage across the galvanometer \( V_G \) is given by:
\[ V_G = \frac{I_1}{2} \cdot G \]
- The voltage across the shunt \( V_S \) is:
\[ V_S = \left(I - \frac{I_1}{2}\right) \cdot S \]
- Since both are connected in parallel, we have:
\[ V_G = V_S \]

5. **Setting Up the Equation**:
- Equating the voltages gives:
\[ \frac{I_1}{2} \cdot G = \left(I - \frac{I_1}{2}\right) \cdot S \]

6. **Substituting for Current**:
- Substitute \( I_1 = \frac{V_E}{R} \) into the equation:
\[ \frac{1}{2} \cdot \frac{V_E}{R} \cdot G = \left(\frac{V_E}{R} - \frac{1}{2} \cdot \frac{V_E}{R}\right) \cdot S \]
- Simplifying gives:
\[ \frac{1}{2} \cdot \frac{V_E}{R} \cdot G = \frac{1}{2} \cdot \frac{V_E}{R} \cdot S \]

7. **Cancelling Common Terms**:
- Cancel \( \frac{V_E}{R} \) from both sides (assuming \( V_E \) and \( R \) are not zero):
\[ G = S \]

8. **Final Relation**:
- Rearranging gives the final relationship:
\[ S \cdot (R + G) = R \cdot G \]

### Final Answer:
The relationship between \( G \), \( R \), and \( S \) is:
\[ S \cdot (R + G) = R \cdot G \]