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Step-by-Step Solution
Step 1: Understand the Problem
We have a battery of emf 20 V and internal resistance 10 Ω. A set of n identical resistors, each of resistance 10 Ω, is first connected in series with the battery. The current in that arrangement is noted as Is. Then the same n resistors are connected in parallel to the same battery, and the resulting current, Ip, is found to be 20 times the previously measured current. We need to determine the value of n.
Step 2: Write the Equivalent Resistance in Series
When the n resistors (each of 10 Ω) are connected in series, the total (external) resistance is
R_{\text{series}} = n \times 10 = 10n\,\Omega.
The total circuit resistance is then:
R_{\text{total, series}} = 10n + 10,
where 10 Ω is the internal resistance of the battery.
Step 3: Current in the Series Arrangement
By Ohm’s law, the current in the series arrangement is
I_{s} = \frac{\text{emf}}{R_{\text{total, series}}} = \frac{20}{10n + 10}.
Simplify by factoring out 10:
I_{s} = \frac{20}{10(1 + n)} = \frac{2}{1 + n}.
Step 4: Write the Equivalent Resistance in Parallel
When the n resistors are connected in parallel, the equivalent (external) resistance becomes
R_{\text{parallel}} = \frac{10}{n}\,\Omega.
Hence, the total circuit resistance now is
R_{\text{total, parallel}} = \frac{10}{n} + 10.
Step 5: Current in the Parallel Arrangement
The current in the parallel arrangement is given by
I_{p} = \frac{\text{emf}}{R_{\text{total, parallel}}} = \frac{20}{\frac{10}{n} + 10}.
We can factor out 10 to simplify:
I_{p} = \frac{20}{10 \left(\frac{1}{n} + 1\right)} = \frac{20}{10 \left(\frac{n + 1}{n}\right)} = \frac{20n}{10(n+1)} = \frac{2n}{n + 1}.
Step 6: Use the Given Ratio of Currents
We know that
I_{p} = 20 \times I_{s}.
So, substitute the expressions for I_{p} and I_{s} :
\[
\frac{2n}{n + 1} = 20 \times \frac{2}{n + 1}.
\]
Step 7: Solve for n
Divide both sides to simplify:
\[
\frac{\frac{2n}{n + 1}}{\frac{2}{n + 1}} = 20.
\]
This becomes
\[
\frac{2n}{n + 1} \times \frac{n + 1}{2} = 20 \quad\Rightarrow\quad n = 20.
\]
Final Answer
The required value of n is 20.
