3.11. CELLS IN SERIES AND IN PARALLEL

Like resistors, cells can be combined together in an electric circuit. And like resistors, one can, for calculating currents and voltages in a circuit, replace a combination of cells by an equivalent cell.

Consider first two cells in series, where one terminal of the two cells is joined together leaving the other terminal in either cell free. Let e₁, e₂ be the emf’s of the two cells and r₁, r₂ their internal resistances, respectively. The potential difference between the terminals is also calculated based on the current flowing through the cells.

When we connect two cells in series, we effectively combine their electromotive forces (emf) and internal resistances to create a single equivalent cell. The positive terminal of one cell connects to the negative terminal of the second, creating a continuous path for current to flow. The total emf of the series combination is simply the sum of their individual emf’s, while the total internal resistance adds up similarly. This leads to the formula:

• Total equivalent emf (e_eq) = e₁ + e₂
• Total equivalent resistance (r_eq) = r₁ + r₂

This means that by understanding the properties of cells in series, we can simplify complex circuits into simpler versions where calculations become easier.

Think of cells in series like a group of people passing water buckets down a line to fill a large container. Each person represents a cell with their own ability to carry water (emf), and the water can only flow through one person at a time (internal resistance). The more people (cells) you add, the more water (current) you can ultimately deliver to the container.

The potential difference between the terminals A and C of the combination is: V_AC = e₁ + e₂ - I(r₁ + r₂)

The formula provided shows how to calculate the total potential difference (voltage) across terminals when two cells in series are connected with a load. Each cell's emf contributes positively to the total voltage, while the current flowing through the internal resistances (r₁ and r₂) creates a voltage drop, thereby reducing the effective voltage available across the load. Hence, to determine the voltage across the external circuit connected to the cells, you add the emf’s and then subtract the voltage drop caused by the internal resistances from the total current I.

Imagine that each cell is like a fountain, where each fountain contributes its water flow (emf) to a main stream (total voltage). But there is a dirt path leading to the stream (internal resistance), which slows down the flow of water. The water that arrives at the stream is less than what the fountains could potentially provide because some is lost along the way—it’s the same with electricity.

Next, consider a parallel combination of the cells. For connections across A and C, the currents I₁ and I₂ flow out of the positive electrodes, with the total current I flowing into the circuit. Since the potential difference across all cells in parallel is the same, we can write: e_eq = e₁ = e₂ And for the internal resistances: 1/r_eq = 1/r₁ + 1/r₂

In a parallel configuration, all the cells share the same voltage across their terminals, which means their emf's remain equal. Unlike in series, the currents from each cell contribute to a total current going through the circuit. The formula for calculating the equivalent internal resistance demonstrates how each cell's resistance decreases the overall resistance of the circuit. When cells are connected in parallel, the combined resistance is less than that of the lowest individual resistance.

Think of parallel cells like several water hoses connected together at the same end. Each hose (cell) can deliver water independently into the same basin (circuit), which means that if one hose is blocked, the others can still deliver water. Because multiple supplies are combined, the overall water flow (current) increases, but since they share the same entrance (voltage), it's not doubled but instead divided among each pathway.

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