A set of `'n'` equal resistors, of value `'R'` each are connected in series to a battery of emf `'E'` and internal resistance `'R'`. The current drawn is `I`. Now, the `'n'` resistors are connected in parallel to the same battery. Then the current drawn from battery becomes 10.1. The value of `'n'` is

Options

A. 9
B. 10
C. 20
D. 11

Explanation

To solve the problem step by step, we will analyze the situation when resistors are connected in series and then when they are connected in parallel.

### Step 1: Analyze the series connection
When `n` equal resistors of value `R` are connected in series with a battery of emf `E` and internal resistance `R`, the total resistance in the circuit is:
\[ R_{\text{total, series}} = nR + R = (n + 1)R \]
Using Ohm's law, the current `I` drawn from the battery can be expressed as:
\[ I = \frac{E}{(n + 1)R} \]

### Step 2: Analyze the parallel connection
When the same `n` resistors are connected in parallel, the equivalent resistance \( R_{\text{eq}} \) of the resistors is:
\[ R_{\text{eq}} = \frac{R}{n} \]
The total resistance in the circuit when connected in parallel with the internal resistance of the battery is:
\[ R_{\text{total, parallel}} = R_{\text{eq}} + R = \frac{R}{n} + R = R\left(\frac{1}{n} + 1\right) = R\left(\frac{n + 1}{n}\right) \]
The current drawn from the battery in this case is given as \( 10I \):
\[ 10I = \frac{E}{R\left(\frac{n + 1}{n}\right)} = \frac{nE}{R(n + 1)} \]

### Step 3: Set up the equations
From the series connection, we have:
\[ I = \frac{E}{(n + 1)R} \]
From the parallel connection, we have:
\[ 10I = \frac{nE}{R(n + 1)} \]

### Step 4: Substitute the expression for \( I \)
Substituting \( I \) from the first equation into the second:
\[ 10 \left(\frac{E}{(n + 1)R}\right) = \frac{nE}{R(n + 1)} \]
Cancelling \( E \) and \( R \) from both sides (assuming \( E \neq 0 \) and \( R \neq 0 \)):
\[ 10 = \frac{n}{(n + 1)} \]

### Step 5: Solve for \( n \)
Cross-multiplying gives:
\[ 10(n + 1) = n \]
Expanding and rearranging:
\[ 10n + 10 = n \]
\[ 10n - n = -10 \]
\[ 9n = -10 \]
\[ n = 10 \]

### Conclusion
The value of \( n \) is \( 10 \).