Let \( E \) be the equivalent electromotive force (EMF) and \( r \) be the equivalent internal resistance of the two cells connected in parallel. The terminal voltages across both cells are equal. This can be expressed as:
\[ E_1 - I_1 r_1 = E_2 - I_2 r_2 = V \]
Here, \( V \) denotes the terminal voltage, \( I_1 \) and \( I_2 \) are the currents through each cell, and \( r_1 \) and \( r_2 \) are their respective internal resistances.
The total current \( I \) from the equivalent cell is the sum of the individual currents:
\[ I = I_1 + I_2 \]
Expressing \( I_1 \) and \( I_2 \) in terms of \( V \):
\[ I_1 = \frac{E_1 - V}{r_1}, \quad I_2 = \frac{E_2 - V}{r_2} \]
Substituting these into the total current equation:
\[ I = \frac{E_1 - V}{r_1} + \frac{E_2 - V}{r_2} \]
Rearranging to express \( I \) as a function of \( V \):
\[ I = \frac{E_1}{r_1} + \frac{E_2}{r_2} - V \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \]
Applying Ohm's law to the equivalent cell, \( V = E - I r \). Substituting the expression for \( I \):
\[ V = E - r \left( \frac{E_1}{r_1} + \frac{E_2}{r_2} - V \left( \frac{1}{r_1} + \frac{1}{r_2} \right) \right) \]
Solving this equation for \( E \) and \( r \) yields:
The equivalent EMF is calculated as:
\[ E = \frac{\frac{E_1}{r_1} + \frac{E_2}{r_2}}{\frac{1}{r_1} + \frac{1}{r_2}} \]
The reciprocal of the equivalent internal resistance is:
\[ \frac{1}{r} = \frac{1}{r_1} + \frac{1}{r_2} \]
Therefore, the equivalent internal resistance is \( r = \frac{r_1 r_2}{r_1 + r_2} \).
Summary of derived values: