Question

Why is the emf of a battery always slightly higher than the potential difference that it can provide to an electric circuit?

Answer 1

The electromotive force (emf) of a battery is indeed slightly higher than the potential difference (or voltage) it can provide to an electric circuit. This discrepancy arises due to the internal resistance of the battery. Here's a detailed explanation of the concepts involved:

### 1. **Understanding EMF and Potential Difference:**
- **Electromotive Force (EMF):** The emf of a battery is a measure of the energy provided per coulomb of charge by the battery’s chemical reactions. It represents the maximum potential difference the battery can supply when no current is flowing (open circuit condition).
  
- **Potential Difference:** When the battery is connected to an external circuit and current flows, the voltage measured across the terminals of the battery is called the potential difference. This is the voltage that actually drives the current through the circuit.

### 2. **Internal Resistance:**
Batteries have an internal resistance, denoted as \( r \). This internal resistance arises from the materials and design of the battery and affects how effectively the battery can deliver its energy to an external circuit.

### 3. **Voltage Drop Across Internal Resistance:**
When a battery supplies current to a circuit, some of the energy from the battery is lost due to the internal resistance. This loss appears as a voltage drop within the battery itself.

The voltage drop across the internal resistance is given by Ohm's law:

\[ V_{drop} = I \cdot r \]

where \( I \) is the current flowing through the circuit and \( r \) is the internal resistance of the battery.

### 4. **Relation Between EMF and Potential Difference:**
The potential difference \( V_{circuit} \) across the terminals of the battery when current \( I \) is flowing is less than the emf \( \mathcal{E} \) of the battery due to the voltage drop across the internal resistance. The relationship can be expressed as:

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

Here:
- \( \mathcal{E} \) is the emf of the battery,
- \( I \) is the current flowing through the external circuit,
- \( r \) is the internal resistance of the battery.

### 5. **Practical Implications:**
- When a battery is not connected to any load (open circuit), the potential difference across its terminals is equal to its emf.
- When the battery is connected to a load and current flows, the potential difference across the terminals will be less than the emf due to the internal voltage drop.

In summary, the emf of a battery is always slightly higher than the potential difference it provides to an external circuit because of the internal resistance, which causes a voltage drop when current flows.

Answer 2

To understand why the electromotive force (EMF) of a battery is always slightly higher than the potential difference it provides to a circuit, we need to delve into the concepts of EMF, internal resistance, and how they interact within a battery.

### 1. **Electromotive Force (EMF)**

- **Definition**: The EMF of a battery is the maximum voltage the battery can provide when no current is flowing. It represents the energy per unit charge supplied by the battery's chemical reactions.
- **Source**: EMF is generated by the chemical reactions within the battery that create a separation of charge between the battery's terminals.

### 2. **Potential Difference**

- **Definition**: The potential difference (also known as terminal voltage) is the voltage you actually measure across the battery's terminals when current is flowing. This is the voltage available to an external circuit.
- **Effect of Current**: When a current flows through the battery, it has to pass through the battery's internal resistance. This internal resistance causes a voltage drop within the battery itself.

### 3. **Internal Resistance**

- **Definition**: Internal resistance is the opposition to the flow of current within the battery. It's due to the resistive components and the electrolyte inside the battery.
- **Impact on Voltage**: According to Ohm’s Law, the voltage drop across the internal resistance is given by \( V_{drop} = I \times R_{internal} \), where \( I \) is the current and \( R_{internal} \) is the internal resistance.

### 4. **Relation Between EMF and Potential Difference**

When a battery is connected to a circuit, the current flowing through the circuit also flows through the internal resistance of the battery. The internal resistance causes a voltage drop that reduces the voltage available across the terminals of the battery. This can be expressed by the formula:

\[ V_{terminal} = EMF - (I \times R_{internal}) \]

Here’s what happens step-by-step:
- **Without Current**: When no current is flowing, \( I = 0 \), so the potential difference across the terminals equals the EMF.
- **With Current**: When current flows, the internal resistance causes a voltage drop. This means the potential difference (terminal voltage) is less than the EMF.

### 5. **Why EMF is Always Higher**

Because the internal resistance causes a voltage drop when current flows, the potential difference (terminal voltage) will always be less than the EMF. The EMF represents the maximum potential difference a battery can provide, but due to the internal resistance, the actual voltage available to an external circuit is slightly reduced.

### **Example for Clarity**

Imagine a battery with an EMF of 12V and an internal resistance of 1Ω. If the battery is supplying a current of 2A to a circuit, the voltage drop across the internal resistance is:

\[ V_{drop} = I \times R_{internal} = 2A \times 1Ω = 2V \]

Thus, the potential difference across the terminals of the battery when delivering 2A is:

\[ V_{terminal} = EMF - V_{drop} = 12V - 2V = 10V \]

In this example, the potential difference (10V) is less than the EMF (12V) due to the voltage drop caused by the internal resistance.

In summary, the EMF of a battery is always slightly higher than the potential difference it can provide to a circuit due to the internal resistance of the battery, which causes a voltage drop when current flows.