Question

Under what condition the terminal potential difference across a battery and it's emf are equal?

Answer 1

The terminal potential difference across a battery (often represented as \( V \)) and its electromotive force (emf, represented as \( \mathcal{E} \)) are equal under specific conditions. Here's a detailed explanation:

### Definitions

1. **Emf (\( \mathcal{E} \))**: 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.

2. **Terminal Potential Difference (\( V \))**: This is the actual voltage output of the battery when it is connected to a circuit and current is flowing. It can be affected by the internal resistance of the battery.

### Conditions for Equality

The terminal potential difference equals the emf when:

1. **No Current is Flowing**:
   - This condition occurs when the battery is open-circuited (not connected to any load). In this case, there is no current flowing through the battery, so there are no voltage drops across any internal resistance. Thus:
     \[
     V = \mathcal{E}
     \]

2. **Internal Resistance is Negligible**:
   - If the internal resistance (\( r \)) of the battery is extremely small compared to the load resistance (\( R \)) in a circuit, and if the current flowing through the circuit is minimal, then the voltage drop across the internal resistance can be negligible. In practical terms:
     \[
     V = \mathcal{E} - I \cdot r \quad \text{(where \( I \) is the current)}
     \]
     If \( I \) is very small, the term \( I \cdot r \) approaches zero, and thus:
     \[
     V \approx \mathcal{E}
     \]

### Summary

To summarize, the terminal potential difference across a battery and its emf are equal when:

- **No current flows** through the battery (open circuit).
- **The internal resistance is negligible**, or the current drawn from the battery is very low, allowing the voltage drop across the internal resistance to be minimal.

These conditions highlight the importance of considering both the circuit configuration and the characteristics of the battery when analyzing voltage in electrical systems.

Answer 2

The terminal potential difference across a battery and its electromotive force (EMF) are equal under the following condition:

**When no current is flowing through the battery.**

Here's a more detailed explanation:

- **EMF (Electromotive Force):** The EMF of a battery is the maximum potential difference it can provide when no current is drawn from it. It represents the total energy per unit charge provided by the battery's chemical reaction.

- **Terminal Potential Difference:** This is the actual voltage measured across the terminals of the battery when a current is flowing. It can be lower than the EMF due to internal resistance in the battery.

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

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

When no current flows (\(I = 0\)), the term \(I \cdot r\) drops out:

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

So, under the condition of no current flowing through the battery, the terminal potential difference across the battery is equal to its EMF.