Understanding Cell EMF and Terminal Voltage

In an electric circuit, a cell provides energy to push charge. The electromotive force, or emf (represented by $\varepsilon$), is the total energy per unit charge supplied by the cell. It's essentially the potential difference across the cell's terminals when no current is flowing.

However, when a cell is connected in a closed circuit and current ($I$) flows through it, the cell itself has an internal resistance ($r$). This internal resistance causes a drop in potential *within* the cell.

The terminal potential difference ($V$) is the actual potential difference measured across the terminals of the cell when current is flowing. This is the voltage available to the external circuit.

Deriving the Relationship

When current ($I$) flows out of the positive terminal of the cell, it passes through the internal resistance ($r$). According to Ohm's Law, the potential drop across this internal resistance is given by:

Potential Drop across internal resistance = $I \times r$

The terminal potential difference ($V$) is the emf ($\varepsilon$) minus the potential drop that occurs within the cell due to its internal resistance. Therefore, the correct relation is:

$V = \varepsilon - Ir$

This equation shows that the terminal voltage ($V$) is always less than the emf ($\varepsilon$) when the cell is discharging (i.e., when current $I$ is flowing out of the positive terminal).

Analyzing the Options

• Option 1: $V = \varepsilon + Ir$. This formula applies when the cell is being *charged* (current flowing into the positive terminal), not when it's used in a closed circuit delivering current.
• Option 2: $V = \varepsilon - Ir$. This correctly represents the terminal potential difference when the cell is discharging, accounting for the voltage drop across the internal resistance.
• Option 3: $V = \varepsilon \times Ir$. This is dimensionally incorrect and does not represent the physical relationship.
• Option 4: $V = \varepsilon / Ir$. This is also dimensionally incorrect and does not represent the physical relationship.

Conclusion

The relationship between the emf ($\varepsilon$), internal resistance ($r$), current ($I$), and terminal potential difference ($V$) for a cell in a closed circuit (discharging) is $V = \varepsilon - Ir$.