Question

Under what condition terminal potential difference is less than emf?

Answer 1

The terminal potential difference (V) of a power source (like a battery) is less than the electromotive force (emf, \( \mathcal{E} \)) when the power source is supplying current to an external circuit and there is a voltage drop across its internal resistance.

Here’s a detailed explanation:

### Understanding EMF and Terminal Potential Difference

1. **Electromotive Force (EMF, \( \mathcal{E} \))**:
   - This is the maximum potential difference that a power source can provide when no current is flowing through it.
   - It represents the work done per unit charge by the power source in moving charges from one terminal to the other.

2. **Terminal Potential Difference (V)**:
   - This is the actual potential difference between the terminals of the power source when it is connected to an external circuit and current is flowing.
   - It can be measured directly across the terminals of the power source.

### Relationship Between EMF and Terminal Potential Difference

When a current \( I \) flows through a power source with an internal resistance \( r \), the terminal potential difference \( V \) is given by:

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

Here’s what’s happening in this equation:

- **EMF (\( \mathcal{E} \))** is the total potential difference provided by the power source.
- **Internal Resistance (\( r \))** is the resistance within the power source itself.
- **Current (\( I \))** is the amount of current flowing through the circuit.

### Condition for Terminal Potential Difference to be Less than EMF

For the terminal potential difference to be less than the EMF, the following condition must be met:

\[ V < \mathcal{E} \]

This inequality holds when the power source is supplying current, meaning \( I \neq 0 \). Specifically, if the current \( I \) is flowing through the internal resistance \( r \), then:

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

Since \( I \cdot r \) is positive (as both \( I \) and \( r \) are positive quantities in practical scenarios), it follows that:

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

### Example

Consider a battery with an EMF of 12 V and an internal resistance of 1 Ω. If the battery is connected to a circuit and current flowing through it is 2 A, the terminal potential difference can be calculated as:

\[ V = \mathcal{E} - I \cdot r = 12\, \text{V} - (2\, \text{A} \times 1\, \Omega) = 12\, \text{V} - 2\, \text{V} = 10\, \text{V} \]

In this case, the terminal potential difference (10 V) is indeed less than the EMF (12 V).

### Conclusion

The terminal potential difference is less than the EMF whenever current is flowing through the power source, due to the voltage drop across its internal resistance. This difference becomes more pronounced with higher currents or larger internal resistances.

Answer 2

The terminal potential difference (V) is less than the electromotive force (emf, E) of a source under the following conditions:

1. **Internal Resistance:** When there is internal resistance (r) in the source, the terminal voltage decreases due to the voltage drop across the internal resistance. The relationship can be expressed as:
   \[
   V = E - Ir
   \]
   where \(I\) is the current flowing from the source.

2. **Load Connected:** When a load is connected to the circuit and draws current, the terminal potential difference will be lower than the emf due to the internal resistance of the source.

3. **Discharge Condition:** In the case of batteries or cells, when they are discharging, the terminal potential difference is often lower than the emf due to the internal resistance.

In summary, as long as there is current flowing and internal resistance present, the terminal potential difference will be less than the emf.