N batteries are connected in parallel, with their terminals...

$N$ batteries are connected in parallel, with their terminals oriented in the same direction. If the $i$ th battery has an emf $\varepsilon_i$ and internal resistance $r_i$ and if the entire set-up can be reduced to a single equivalent battery with emf $\varepsilon_{e q}$ and internal resistance $r_{e q}$ with respect to two terminals at the ends of a parallel branch, determine $\varepsilon_{e q}$ and $r_{e q}$.

Key Concepts

Open-Circuit Voltage Calculation

The open-circuit voltage is the voltage measured across the terminals of a network when no load is connected. For a parallel battery setup, it requires balancing the contributions from each battery's emf based on its internal resistance. Essentially, the equivalent emf is a weighted average of the individual emfs, with weights inversely proportional to the internal resistances, ensuring that the total current in the circuit is zero under open-circuit conditions.

Parallel Combination of Resistances

When resistors (or internal resistances in this context) are connected in parallel, the total equivalent resistance is given by the reciprocal of the sum of the reciprocals of each individual resistance. This principle is crucial when determining the effective internal resistance of batteries connected in parallel, as it results in an equivalent resistance that is lower than any of the individual resistances.

Parallel Connection of Voltage Sources

This concept refers to connecting multiple voltage sources together such that their corresponding terminals (positive to positive and negative to negative) are linked. When doing so, especially if the sources have different electromotive forces (emfs) and internal resistances, current can circulate between the sources, and the overall behavior of the network needs careful analysis. This concept is fundamental for understanding how to combine sources to form a single equivalent source.

Equivalent Circuit (Thevenin’s Theorem)

Thevenin’s theorem allows us to simplify a complex linear circuit into a single voltage source in series with a single resistor. In the case of batteries connected in parallel, the network can be reduced to one equivalent battery by determining an overall emf and combined internal resistance that will behave identically to the original network when any load is connected. This makes analyzing the circuit much simpler.

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