A circuit consists of a resistance R connected to n similar cells. If the current in the circuit is the same whether the cells are connected in series or in parallel, then the internal resistance r of each cell is given by
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( r=R \)
For \( n \) identical cells of emf \( \varepsilon \) and internal resistance \( r \), the equivalent cell depends on how they are joined.
A series combination has emf \( n\varepsilon \) and internal resistance \( nr \).
A parallel combination has emf \( \varepsilon \) and internal resistance \( r/n \).
In each case the current is the equivalent emf divided by the total resistance (external plus internal).
With the cells in series, the current is:
\( \displaystyle I_s=\frac{n\varepsilon}{R+nr} \)
With the cells in parallel, it is:
\( \begin{aligned} I_p &= \frac{\varepsilon}{R+r/n} \\ &= \frac{n\varepsilon}{nR+r} \end{aligned} \)
Equating the two currents:
\( \displaystyle \frac{n\varepsilon}{R+nr}=\frac{n\varepsilon}{nR+r} \)
\( \Rightarrow\; R+nr=nR+r \)
\( (n-1)\,r=(n-1)\,R \)
\( \Rightarrow\; r=R \quad (n\gt1) \)