A battery has an emf of 12 V and an internal resistance of `2 Omega`. Is the terminal-to-terminal potential difference greatest then less than, or equal to 12V if the current in the battery is (a) from the negative to the positive terminal, (b) from the positive to the negative terminal, and (c) zero?

Answer

To solve the problem, we need to analyze the terminal-to-terminal potential difference (V) of a battery with an electromotive force (emf, E) of 12 V and an internal resistance (r) of 2 Ω under different current conditions.

### Step-by-Step Solution:

1. **Understanding the Battery Parameters**:
- Given:
- EMF (E) = 12 V
- Internal Resistance (r) = 2 Ω

2. **Case (a): Current from Negative to Positive Terminal**:
- When the current (I) flows from the negative terminal to the positive terminal, the battery is supplying power.
- The terminal-to-terminal potential difference can be calculated using the formula:
\[ V = E - I \cdot r \]
- Since the current is flowing out of the positive terminal, it causes a voltage drop across the internal resistance.
- Therefore, the terminal potential difference will be less than the EMF:
\[ V < 12 \text{ V} \]

3. **Case (b): Current from Positive to Negative Terminal**:
- When the current flows from the positive terminal to the negative terminal, the battery is being charged.
- In this case, the terminal-to-terminal potential difference can be calculated as:
\[ V = E + I \cdot r \]
- Here, the current adds to the EMF because the battery is being charged, resulting in a potential difference greater than the EMF:
\[ V > 12 \text{ V} \]

4. **Case (c): Current is Zero**:
- When there is no current (I = 0), the terminal-to-terminal potential difference is simply equal to the EMF:
\[ V = E \]
- Thus, in this case:
\[ V = 12 \text{ V} \]

### Summary of Results:
- For (a): Terminal potential difference is **less than 12 V**.
- For (b): Terminal potential difference is **greater than 12 V**.
- For (c): Terminal potential difference is **equal to 12 V**.