Question

Two cells are of emf's \(E_1\) and \(E_2\) and their internal resistances are \(r_1\) and \(r_2\) respectively. They are joined in parallel to each other. Obtain the formula for the equivalent emf of this combination of cells.

Hint:

When cells are connected in parallel, the equivalent emf is a weighted average of the individual emfs, where the internal resistances of the cells act as weights.

Answer 1

When two cells with different electromotive forces (emfs) and internal resistances are connected in parallel, the equivalent emf \(E_{\text{eq}}\) and the equivalent internal resistance \(r_{\text{eq}}\) can be found using the following approach:
Step 1: Calculate the equivalent internal resistance.
The two internal resistances \(r_1\) and \(r_2\) are in parallel, so the total internal resistance \(r_{\text{eq}}\) is given by:
\[ \frac{1}{r_{\text{eq}}} = \frac{1}{r_1} + \frac{1}{r_2} \] Therefore, the equivalent internal resistance is:
\[ r_{\text{eq}} = \frac{r_1 r_2}{r_1 + r_2} \] Step 2: Calculate the equivalent emf.
The equivalent emf of two cells in parallel is given by the formula:
\[ E_{\text{eq}} = \frac{E_1 r_2 + E_2 r_1}{r_1 + r_2} \] This formula takes into account both the emfs and the internal resistances of the two cells. The equivalent emf is a weighted average of the two emfs, where the internal resistances act as the weights.
Thus, the formula for the equivalent emf of the combination of cells is:
\[ E_{\text{eq}} = \frac{E_1 r_2 + E_2 r_1}{r_1 + r_2} \]