Question

Is terminal potential difference equal to the emf of a cell justify your answer?

Answer 1

No, the terminal potential difference of a cell is not always equal to the electromotive force (emf) of the cell. The terminal potential difference is only equal to the emf of a cell under certain conditions. Let me explain in detail:

### Definitions:

1. **EMF (Electromotive Force):**
   - EMF is the maximum potential difference between the terminals of a cell when no current is drawn from it (i.e., when the circuit is open).
   - It represents the total energy provided by the cell to move a unit charge through the entire circuit, including both the external circuit and the internal resistance of the cell.
   - The EMF of a cell is constant and depends on the chemical composition of the cell.

2. **Terminal Potential Difference (V):**
   - The terminal potential difference is the voltage across the terminals of the cell when a current is flowing.
   - This value can change based on the amount of current flowing and the internal resistance of the cell.

### Internal Resistance of the Cell:
- Every real battery or cell has some internal resistance, denoted by \( r \). This internal resistance affects the terminal voltage when a current flows through the circuit.

### Relationship between EMF, Terminal Potential Difference, and Internal Resistance:
The relationship between the terminal potential difference \( V \), the EMF \( \mathcal{E} \), the current \( I \), and the internal resistance \( r \) of the cell is given by the formula:
\[
V = \mathcal{E} - Ir
\]
Where:
- \( \mathcal{E} \) is the EMF of the cell.
- \( V \) is the terminal potential difference.
- \( I \) is the current flowing through the circuit.
- \( r \) is the internal resistance of the cell.

### Justification:

1. **When No Current is Drawn (Open Circuit):**
   - If the cell is not connected to a load and no current is flowing (i.e., \( I = 0 \)), the terminal potential difference is equal to the EMF of the cell.
   - In this case, \( V = \mathcal{E} \) because there is no voltage drop across the internal resistance since \( Ir = 0 \).

2. **When Current is Drawn (Closed Circuit):**
   - When a current flows through the circuit (i.e., \( I > 0 \)), there is a voltage drop across the internal resistance of the cell, which reduces the terminal potential difference.
   - The terminal potential difference \( V \) becomes less than the EMF by an amount equal to the voltage drop across the internal resistance (\( Ir \)).
   - Thus, \( V = \mathcal{E} - Ir \), meaning the terminal potential difference is lower than the EMF.

### Conclusion:
- **The terminal potential difference is only equal to the EMF when no current flows through the cell (open circuit).**
- **When a current flows, the terminal potential difference is less than the EMF due to the internal resistance of the cell.** The more current that flows, the greater the voltage drop across the internal resistance, further lowering the terminal potential difference.

Answer 2

The terminal potential difference (often just called the terminal voltage) of a cell is not always equal to its electromotive force (emf). Here’s a detailed explanation to justify this statement:

### Understanding Key Concepts

1. **Electromotive Force (emf):**
   - The emf of a cell is the maximum potential difference between the cell’s terminals when no current is flowing. It represents the energy provided per unit charge by the cell's internal chemical reactions.

2. **Terminal Potential Difference (Terminal Voltage):**
   - The terminal potential difference is the actual voltage measured across the cell’s terminals when current is flowing through the cell. This value can be different from the emf due to internal factors within the cell.

### Factors Affecting Terminal Potential Difference

1. **Internal Resistance:**
   - Every cell has some internal resistance (r). When a current (I) flows through the cell, there is a voltage drop across this internal resistance, described by Ohm’s Law: \( V_{drop} = I \times r \).

2. **Relationship Between emf and Terminal Voltage:**
   - When a current flows, the terminal voltage (V) is given by:
     \[
     V = \text{emf} - I \times r
     \]
   - This equation shows that the terminal voltage is reduced by the voltage drop due to the internal resistance of the cell. Therefore, the terminal voltage will always be less than the emf when current is flowing.

### Example Illustration

1. **No Current (Open Circuit Condition):**
   - If no current is flowing (open circuit), the terminal potential difference equals the emf because there is no internal voltage drop.
   - For instance, if a cell has an emf of 1.5 volts and no current is flowing, the terminal voltage is 1.5 volts.

2. **Current Flowing (Closed Circuit Condition):**
   - When the cell is connected to a circuit and current flows, the internal resistance causes a voltage drop. If the cell has an internal resistance of 0.5 ohms and is supplying a current of 2 amps, the voltage drop across the internal resistance is:
     \[
     V_{drop} = I \times r = 2 \text{ A} \times 0.5 \text{ } \Omega = 1 \text{ V}
     \]
   - If the emf of the cell is 1.5 volts, the terminal voltage will be:
     \[
     V = \text{emf} - V_{drop} = 1.5 \text{ V} - 1 \text{ V} = 0.5 \text{ V}
     \]

### Summary

- **When no current is flowing**, the terminal potential difference equals the emf of the cell.
- **When current is flowing**, the terminal potential difference is less than the emf due to the voltage drop across the internal resistance.

Therefore, the terminal potential difference and the emf are equal only under open circuit conditions. In practical scenarios where current flows, they differ, with the terminal voltage being lower than the emf.