The electromotive force (emf) of a cell is always greater than its terminal potential difference due to the internal resistance of the cell. Here's a detailed explanation:
### **Definitions**
1. **Electromotive Force (emf):** This is the maximum potential difference that a cell or battery can provide when no current is flowing through the circuit. It's the measure of the energy per unit charge provided by the cell’s chemical reactions.
2. **Terminal Potential Difference (TPD):** This is the potential difference across the terminals of the cell when it is supplying current to the external circuit. It is the voltage you measure across the cell's terminals when the circuit is closed and current flows.
### **Why Emf is Greater Than Terminal Potential Difference**
When a cell is connected to a circuit, it supplies current, and the following factors come into play:
1. **Internal Resistance (r):** Every real cell has some internal resistance due to the materials and construction of the cell. This resistance causes a voltage drop inside the cell as current flows.
2. **Current (I):** When current flows through the cell, a voltage drop occurs across the internal resistance of the cell. This can be described using Ohm's law: \( V = IR \), where \( R \) is the resistance and \( I \) is the current.
### **Mathematical Explanation**
The relationship between emf (\( \mathcal{E} \)), terminal potential difference (V), internal resistance (r), and current (I) can be described by the following equations:
1. **Emf Equation:**
\[ \mathcal{E} = V + Ir \]
Here, \( \mathcal{E} \) is the emf, \( V \) is the terminal potential difference, \( I \) is the current, and \( r \) is the internal resistance.
2. **Terminal Potential Difference:**
\[ V = \mathcal{E} - Ir \]
This equation shows that the terminal potential difference is less than the emf by the amount of the voltage drop caused by the internal resistance.
### **Explanation**
- **No Current Flow:** When no current flows (open circuit), the terminal potential difference equals the emf, as there’s no internal voltage drop.
- **Current Flow:** When current flows (closed circuit), the internal resistance causes a voltage drop, reducing the terminal potential difference below the emf. Specifically, the terminal potential difference is the emf minus the voltage drop across the internal resistance: \( V = \mathcal{E} - Ir \).
### **Summary**
The emf of a cell represents its maximum potential difference when not under load, while the terminal potential difference is the actual voltage available when current is drawn. The difference arises due to the internal resistance of the cell, which causes a voltage drop as current flows. Thus, the emf is always greater than the terminal potential difference in practical situations.