A voltmeter has range 0 `to` V with a series resistance R. With a series resistance 2R, the range is 0 `to` V'. The correct relation between v and v' is

Options

A. `V' = 2V`
B. `V' gt 2V`
C. `V' lt 2V`
D. `V' gt gt 2V`

Explanation

To solve the problem, we need to establish the relationship between the voltages \( V \) and \( V' \) when the voltmeter is connected with different series resistances.

### Step-by-step Solution:

1. **Understanding the Setup**:
- A voltmeter with a range of \( 0 \) to \( V \) has a series resistance \( R \).
- When the series resistance is changed to \( 2R \), the new range of the voltmeter is \( 0 \) to \( V' \).

2. **Current Calculation for the First Setup**:
- The current \( I \) flowing through the circuit when the voltmeter is connected with resistance \( R \) can be expressed using Ohm's Law:
\[ I = \frac{V}{R + R_v} \]
- Here, \( R_v \) is the internal resistance of the voltmeter.

3. **Voltage Reading for the First Setup**:
- The voltage reading on the voltmeter, \( V \), can be expressed as:
\[ V = I \cdot R_v = \frac{V}{R + R_v} \cdot R_v \]

4. **Current Calculation for the Second Setup**:
- In the second setup, where the series resistance is \( 2R \), the current \( I' \) can be calculated as:
\[ I' = \frac{V'}{2R + R_v} \]

5. **Voltage Reading for the Second Setup**:
- The voltage reading on the voltmeter in this case, \( V' \), is given by:
\[ V' = I' \cdot R_v = \frac{V'}{2R + R_v} \cdot R_v \]

6. **Setting Up the Equations**:
- From the first setup, we have:
\[ V = \frac{V \cdot R_v}{R + R_v} \]
- From the second setup:
\[ V' = \frac{V' \cdot R_v}{2R + R_v} \]

7. **Relating \( V \) and \( V' \)**:
- Since both expressions for \( V \) and \( V' \) involve the same \( R_v \), we can set up the relationship:
\[ \frac{V}{R + R_v} = \frac{V'}{2R + R_v} \]

8. **Cross Multiplying to Solve for \( V' \)**:
- Cross-multiplying gives:
\[ V(2R + R_v) = V'(R + R_v) \]
- Rearranging this leads to:
\[ V' = \frac{V(2R + R_v)}{R + R_v} \]

9. **Simplifying the Expression**:
- Further simplification leads us to:
\[ V' = 2V \cdot \frac{R + R_v - R_v}{R + R_v} = 2V \cdot \frac{R}{R + R_v} \]

10. **Conclusion**:
- Thus, the relationship between \( V \) and \( V' \) can be summarized as:
\[ V' = 2V \cdot \frac{R}{R + R_v} \]