Question

What is the relationship between terminal potential difference and EMF?

Answer 1

The relationship between terminal potential difference (V) and electromotive force (EMF, \( \mathcal{E} \)) is fundamental in understanding how electrical circuits work. Here’s a detailed explanation:

### Definitions

1. **EMF (Electromotive Force, \( \mathcal{E} \))**:
   - This is the maximum potential difference between the terminals of a power source (such as a battery or generator) when no current is flowing. It represents the energy per unit charge provided by the source to move the charge through the circuit.

2. **Terminal Potential Difference (V)**:
   - This is the actual voltage measured across the terminals of the power source when it is connected to a circuit and current is flowing. It is the potential difference that is available to drive the current through the external circuit.

### Relationship

The relationship between EMF (\( \mathcal{E} \)) and terminal potential difference (V) can be expressed as:

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

where:
- \( I \) is the current flowing through the circuit.
- \( r \) is the internal resistance of the power source.

### Explanation

1. **Internal Resistance**:
   - Power sources like batteries have internal resistance, which causes a drop in the voltage as current flows through the source. This drop is proportional to the current and the internal resistance (\( I \cdot r \)).

2. **When No Current Flows**:
   - If the circuit is open (i.e., no current flows), the terminal potential difference equals the EMF of the source because there is no voltage drop across the internal resistance.

3. **When Current Flows**:
   - When the circuit is closed and current flows, the terminal potential difference is less than the EMF. This is due to the voltage drop across the internal resistance of the power source. The greater the current or the internal resistance, the lower the terminal potential difference compared to the EMF.

### Summary

- **No Current:** \( V = \mathcal{E} \)
- **With Current:** \( V = \mathcal{E} - I \cdot r \)

This relationship is crucial for accurately assessing the performance of electrical circuits and power sources.

Answer 2

The relationship between terminal potential difference (TPD) and electromotive force (EMF) is a key concept in electrical engineering, particularly in the study of electrical circuits and batteries. Here's a detailed explanation:

### Electromotive Force (EMF)

**Definition:**
- The EMF of a source (like a battery or a generator) is the maximum potential difference between its terminals when no current is flowing through the circuit. It represents the work done per unit charge to move the charge from one terminal to the other.

**Characteristics:**
- EMF is essentially the "ideal" voltage of the source, as if there were no internal resistance. It is a measure of the source's ability to drive current through a circuit.

### Terminal Potential Difference (TPD)

**Definition:**
- The terminal potential difference is the actual voltage measured across the terminals of a source when current is flowing. This is the voltage available to the external circuit.

**Characteristics:**
- TPD can be less than the EMF due to the internal resistance of the source.

### Relationship Between EMF and TPD

The relationship between EMF (\( \mathcal{E} \)) and terminal potential difference (V) is affected by the internal resistance (\( r \)) of the source and the current (\( I \)) flowing through the circuit.

**Mathematical Relationship:**
\[ V = \mathcal{E} - I \cdot r \]

- **\( \mathcal{E} \)**: Electromotive Force
- **\( I \)**: Current flowing through the circuit
- **\( r \)**: Internal resistance of the source
- **\( V \)**: Terminal potential difference

### How the Relationship Works

1. **No Current Flowing (Open Circuit):**
   - When there is no current flowing (open circuit condition), the terminal potential difference equals the EMF: \( V = \mathcal{E} \). This is because there is no voltage drop across the internal resistance.

2. **Current Flowing (Closed Circuit):**
   - When current flows, the terminal potential difference is reduced compared to the EMF due to the voltage drop across the internal resistance: \( V = \mathcal{E} - I \cdot r \). Here, \( I \cdot r \) represents the internal voltage drop caused by the internal resistance.

3. **Internal Resistance Impact:**
   - The internal resistance of the source affects how much the terminal potential difference will drop below the EMF. Higher internal resistance leads to a larger voltage drop for the same current, and thus a lower terminal potential difference.

### Practical Example

Consider a battery with an EMF of 12V and an internal resistance of 0.5 ohms. If the current drawn by the circuit is 2A, the terminal potential difference can be calculated as follows:

1. Calculate the voltage drop across the internal resistance:
   \[ I \cdot r = 2A \cdot 0.5\Omega = 1V \]

2. Subtract this drop from the EMF:
   \[ V = \mathcal{E} - I \cdot r = 12V - 1V = 11V \]

So, the terminal potential difference is 11V.

### Summary

- **EMF** is the theoretical maximum potential difference and is determined by the source’s ability to do work.
- **Terminal potential difference** is the actual voltage available for use in the external circuit and is always less than or equal to the EMF due to internal resistance.

Understanding this relationship helps in analyzing and designing electrical circuits, ensuring that the source is used efficiently and effectively.