Question

When the terminal potential difference of a battery is equal to its emf?

Answer 1

When the terminal potential difference of a battery equals its electromotive force (emf), the battery is operating under ideal conditions—specifically, when there is no current flowing through the battery. This situation occurs when the battery is open-circuited, meaning it's not connected to any external load. In this state, the entire voltage produced by the chemical reactions inside the battery is available at its terminals, resulting in the terminal potential difference matching the emf. If the battery is connected to a circuit and supplying current, the terminal voltage will be lower than the emf due to internal resistance. Would you like to delve deeper into how this relates to battery efficiency?

Answer 2

The terminal potential difference of a battery equals its electromotive force (emf) under ideal conditions where no current is flowing. Let me explain this in detail:

### Understanding EMF and Terminal Potential Difference

1. **Electromotive Force (EMF):** This is the maximum potential difference provided by a battery when no current is flowing. It represents the energy per unit charge supplied by the battery's internal chemical processes. EMF is an idealized concept that assumes perfect conditions with no internal resistance.

2. **Terminal Potential Difference:** This is the actual voltage measured across the terminals of the battery when it is connected to a circuit and current is flowing. This potential difference takes into account the internal resistance of the battery.

### Internal Resistance

A real battery has some internal resistance, denoted as \( r \). When a current \( I \) flows through the battery, there is a voltage drop across this internal resistance due to Ohm's Law (\( V = IR \)).

### Relationship Between EMF and Terminal Potential Difference

The relationship between the EMF (\( \mathcal{E} \)), the terminal potential difference (\( V \)), and the internal resistance (\( r \)) is given by:

\[ V = \mathcal{E} - Ir \]

Where:
- \( V \) is the terminal potential difference.
- \( \mathcal{E} \) is the EMF of the battery.
- \( I \) is the current flowing through the circuit.
- \( r \) is the internal resistance of the battery.

### When Terminal Potential Difference Equals EMF

The terminal potential difference equals the EMF of the battery in the following condition:

- **No Current Flowing:** When no current is flowing through the battery, \( I = 0 \). Substituting \( I = 0 \) into the equation:

  \[
  V = \mathcal{E} - Ir
  \]

  This simplifies to:

  \[
  V = \mathcal{E} - 0 \cdot r
  \]

  \[
  V = \mathcal{E}
  \]

In this situation, the terminal potential difference \( V \) is equal to the EMF \( \mathcal{E} \) because there is no voltage drop due to internal resistance.

### Summary

In summary, the terminal potential difference of a battery equals its EMF only when there is no current flowing through the battery. Under normal operating conditions where current is flowing, the terminal potential difference will be less than the EMF due to the voltage drop across the internal resistance of the battery.