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 solve the problem, we need 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. Here are the steps to derive the solution:

### Step 1: Understand the Circuit
We have a cell with an electromotive force (E) and an internal resistance (r). It is connected in series with an external resistance (nr), where n is a constant.

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

### Step 3: Calculate the Current (I)
Using Ohm's Law, the current (I) flowing through the circuit can be calculated using the formula:
\[ I = \frac{E}{R_{\text{total}}} = \frac{E}{r(1 + n)} \]

### Step 4: Calculate the Terminal Potential Difference (V)
The terminal potential difference (V) can be found using the formula:
\[ V = E - I \cdot r \]
Substituting the expression for I:
\[ V = E - \left(\frac{E}{r(1 + n)}\right) \cdot r \]
\[ V = E - \frac{E}{1 + n} \]
\[ V = E \left(1 - \frac{1}{1 + n}\right) \]
\[ V = E \left(\frac{(1 + n) - 1}{1 + n}\right) \]
\[ V = E \left(\frac{n}{1 + n}\right) \]

### Step 5: Calculate the Ratio of Terminal Potential Difference to E.M.F.
Now, we can find the ratio of the terminal potential difference (V) to the electromotive force (E):
\[ \frac{V}{E} = \frac{E \cdot \frac{n}{1 + n}}{E} \]
\[ \frac{V}{E} = \frac{n}{1 + n} \]

### Conclusion
Thus, the ratio of the terminal potential difference to the electromotive force is:
\[ \frac{V}{E} = \frac{n}{n + 1} \]

### Final Answer
The correct answer is \( \frac{n}{n + 1} \).