The maximum safe voltage that can be measured using a galvanometer of resistance G is `V_(m)` . Find the resistance to be connected in series with the galvanometer so that it becomes a voltmeter of range `nV_(m)`.

Answer

To solve the problem of finding the resistance to be connected in series with a galvanometer to convert it into a voltmeter of range \( nV_m \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have a galvanometer with a maximum safe voltage \( V_m \) and resistance \( G \).
- We want to measure a higher voltage, specifically \( nV_m \), using the galvanometer.

2. **Voltage Drop Across the Galvanometer**:
- The galvanometer can safely measure up to \( V_m \). When we want to measure \( nV_m \), the additional voltage that needs to be dropped across the series resistance \( R_s \) is:
\[ V_{drop} = nV_m - V_m = (n - 1)V_m \]

3. **Current Through the Circuit**:
- The current \( I_G \) flowing through the galvanometer when measuring the maximum voltage \( V_m \) can be expressed as:
\[ I_G = \frac{V_m}{G} \]
- This is because the current through the galvanometer is the voltage across it divided by its resistance.

4. **Total Voltage in the Circuit**:
- When the galvanometer and the series resistance \( R_s \) are connected, the total voltage across the circuit is \( nV_m \). The relationship can be expressed as:
\[ I_G = \frac{nV_m}{R_{eq}} \]
- Here, \( R_{eq} \) is the equivalent resistance of the circuit, which is the sum of the galvanometer resistance and the series resistance:
\[ R_{eq} = R_s + G \]

5. **Setting Up the Equation**:
- Since both expressions for \( I_G \) are equal, we can set them equal to each other:
\[ \frac{V_m}{G} = \frac{nV_m}{R_s + G} \]

6. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ V_m(R_s + G) = nV_m \cdot G \]

7. **Simplifying the Equation**:
- Dividing both sides by \( V_m \) (assuming \( V_m \neq 0 \)):
\[ R_s + G = nG \]
- Rearranging gives:
\[ R_s = nG - G = (n - 1)G \]

8. **Final Result**:
- Therefore, the resistance \( R_s \) that needs to be connected in series with the galvanometer is:
\[ R_s = (n - 1)G \]

### Summary:
To convert a galvanometer with resistance \( G \) into a voltmeter capable of measuring up to \( nV_m \), connect a resistance of \( (n-1)G \) in series with it.