A cell of e.mf. E and internal resistance r is connected in series with an external resistance nr. Then, the ratio of the terminal potential difference to E.M.F.is

Options

A. `1//n`
B. `(1)/(n+1)`
C. `(n)/(n=1)`
D. `(n+1)/(n)`

Answer

To find the ratio of the terminal potential difference (V) to the electromotive force (E) of a cell connected in series with an external resistance, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
- Let the EMF of the cell be \( E \).
- Let the internal resistance of the cell be \( r \).
- Let the external resistance be \( nr \), where \( n \) is a constant.

2. **Calculate the Total Resistance**:
- The total resistance \( R_{total} \) in the circuit is the sum of the internal resistance and the external resistance:
\[ R_{total} = r + nr = r(1 + n) \]

3. **Determine the Current in the Circuit**:
- Using Ohm's law, the current \( I \) flowing through the circuit can be calculated as:
\[ I = \frac{E}{R_{total}} = \frac{E}{r(1 + n)} \]

4. **Calculate the Terminal Potential Difference (V)**:
- The terminal potential difference \( V \) can be expressed using the formula:
\[ V = E - I \cdot r \]
- Substituting the value of \( I \):
\[ V = E - \left(\frac{E}{r(1 + n)}\right) \cdot r \]
- Simplifying this:
\[ V = E - \frac{E}{1 + n} \]
- To combine these terms, we can find a common denominator:
\[ V = \frac{E(1 + n) - E}{1 + n} = \frac{En}{1 + n} \]

5. **Find the Ratio of Terminal Potential Difference to EMF**:
- Now, we need to find the ratio \( \frac{V}{E} \):
\[ \frac{V}{E} = \frac{\frac{En}{1 + n}}{E} \]
- This simplifies to:
\[ \frac{V}{E} = \frac{n}{1 + n} \]

### Final Answer:
The ratio of the terminal potential difference to the EMF is:
\[ \frac{V}{E} = \frac{n}{n + 1} \]