Question

Two cells are connected in parallel by connecting their positive terminals together and connecting their negative terminals together. Derive expression for the equivalent emf and the equivalent internal resistance of the combination.

Hint:

In parallel both cells have the same terminal voltage V, and the currents add. Write V = E - I r for each cell and compare with one cell.

Answer 1

Step 1: Set up:
Let the cells have emf \( E_1, E_2 \) and internal resistances \( r_1, r_2 \). Join their positive terminals at A and their negative terminals at B.
Let the cells send currents \( I_1 \) and \( I_2 \) into the external circuit, so the total current is \( I = I_1 + I_2 \).

Step 2: Terminal voltage of each cell:
Both cells are across the same two points, so the voltage \( V = V_A - V_B \) is the same.
\( V = E_1 - I_1 r_1 \) and \( V = E_2 - I_2 r_2 \).
So \( I_1 = \dfrac{E_1 - V}{r_1} \) and \( I_2 = \dfrac{E_2 - V}{r_2} \).

Step 3: Add the currents:
\[ I = I_1 + I_2 = \left(\frac{E_1}{r_1} + \frac{E_2}{r_2}\right) - V\left(\frac{1}{r_1} + \frac{1}{r_2}\right) \]
Solve for V:
\[ V = \frac{E_1 r_2 + E_2 r_1}{r_1 + r_2} - I\,\frac{r_1 r_2}{r_1 + r_2} \]

Step 4: Compare with one cell:
A single cell gives \( V = E_{eq} - I r_{eq} \). So
\[ E_{eq} = \frac{E_1 r_2 + E_2 r_1}{r_1 + r_2}, \qquad r_{eq} = \frac{r_1 r_2}{r_1 + r_2} \]

Final Answer:
\( E_{eq} = \dfrac{E_1 r_2 + E_2 r_1}{r_1 + r_2} \) and \( r_{eq} = \dfrac{r_1 r_2}{r_1 + r_2} \). \[ \boxed{E_{eq} = \frac{E_1 r_2 + E_2 r_1}{r_1 + r_2},\ r_{eq} = \frac{r_1 r_2}{r_1 + r_2}} \]