Question

For two identical cells each having emf \(E\) and internal resistance \(r\), the current through an external resistor of \(6\,\Omega\) is the same when the cells are connected in series as well as in parallel. The value of the internal resistance \(r\) is ________ \(\Omega\).

• A. \(9\)
• B. \(3\)
• C. \(6\)
• D. \(4\)

Hint:

When currents are equal in series and parallel cell combinations, equate the current expressions directly to find internal resistance.

Answer 1

The Correct Option is B

Concept: Current through an external resistance depends on the total emf and total resistance of the circuit. For series and parallel combinations of identical cells, these quantities differ.
Step 1: Current when cells are in series Total emf: \[ E_s = 2E \] Total internal resistance: \[ r_s = 2r \] Current: \[ I_s=\frac{2E}{6+2r} \]
Step 2: Current when cells are in parallel Equivalent emf: \[ E_p=E \] Equivalent internal resistance: \[ r_p=\frac{r}{2} \] Current: \[ I_p=\frac{E}{6+\frac{r}{2}} \]
Step 3: Equate the two currents \[ \frac{2E}{6+2r}=\frac{E}{6+\frac{r}{2}} \] Cancel \(E\): \[ \frac{2}{6+2r}=\frac{1}{6+\frac{r}{2}} \] Cross-multiplying: \[ 2\left(6+\frac{r}{2}\right)=6+2r \] \[ 12+r=6+2r \Rightarrow r=3 \]