Question

Two cells of emf 4V and 2V having respective internal resistance of 1 Ω and 2 Ω are connected in parallel, so as to send current in the same direction through an external resistance of 5 Ω. Find the current through the external resistance.

Hint:

For parallel cells, use equivalent emf and resistance; ensure same direction for current flow.

Answer 1

For parallel cells with emf \( E_1 = 4 \, \text{V} \), \( r_1 = 1 \, \Omega \), \( E_2 = 2 \, \text{V} \), \( r_2 = 2 \, \Omega \), equivalent emf:
\[ E_{\text{eq}} = \frac{E_1 r_2 + E_2 r_1}{r_1 + r_2} = \frac{4 \times 2 + 2 \times 1}{1 + 2} = \frac{8 + 2}{3} = \frac{10}{3} \, \text{V}. \] Equivalent internal resistance:
\[ \frac{1}{r_{\text{eq}}} = \frac{1}{r_1} + \frac{1}{r_2} = \frac{1}{1} + \frac{1}{2} = \frac{3}{2} \quad \Rightarrow \quad r_{\text{eq}} = \frac{2}{3} \, \Omega. \] Total resistance: \( r_{\text{eq}} + R = \frac{2}{3} + 5 = \frac{17}{3} \, \Omega \).
Current through external resistance:
\[ I = \frac{E_{\text{eq}}}{r_{\text{eq}} + R} = \frac{\frac{10}{3}}{\frac{17}{3}} = \frac{10}{17} \, \text{A} \approx 0.588 \, \text{A}. \] Answer: \( \frac{10}{17} \, \text{A} \).