Consider four circuits shown in the figure below. In which circuit power dissipated is greatest (Neglect the internal resistance of the power supply)?

Options

A.
B.
C.
D.

Explanation

To determine which circuit dissipates the greatest power, we can use the formula for power dissipation in a circuit, which is given by Joule's law:

\[ P = \frac{E^2}{R_{eq}} \]

Where:
- \( P \) is the power dissipated,
- \( E \) is the voltage of the power supply (which is constant for all circuits),
- \( R_{eq} \) is the equivalent resistance of the circuit.

Since \( E \) is constant across all circuits, the power dissipation is inversely proportional to the equivalent resistance. Therefore, to find the circuit with the greatest power dissipation, we need to find the circuit with the minimum equivalent resistance.

### Step-by-Step Solution:

1. **Analyze Circuit A:**
- The two resistors are in parallel.
- The formula for equivalent resistance \( R_{eq} \) for two resistors \( R \) in parallel is:
\[ R_{eq} = \frac{R \cdot R}{R + R} = \frac{R^2}{2R} = \frac{R}{2} \]

2. **Analyze Circuit B:**
- The two resistors are in series.
- The equivalent resistance is:
\[ R_{eq} = R + R = 2R \]

3. **Analyze Circuit C:**
- The two resistors are in parallel.
- The equivalent resistance is:
\[ R_{eq} = \frac{R \cdot R}{R + R} = \frac{R^2}{2R} = \frac{R}{2} \]
- However, if we consider the arrangement correctly, if they are in series with another resistor, we might need to recalculate based on the configuration.

4. **Analyze Circuit D:**
- The two resistors are in series, and this combination is in parallel with another resistor.
- The equivalent resistance for the series part is \( 2R \), and now we find the equivalent resistance with the parallel resistor \( R \):
\[ R_{eq} = \frac{2R \cdot R}{2R + R} = \frac{2R^2}{3R} = \frac{2R}{3} \]

5. **Compare Equivalent Resistances:**
- Circuit A: \( R_{eq} = \frac{R}{2} \)
- Circuit B: \( R_{eq} = 2R \)
- Circuit C: \( R_{eq} = \frac{R}{2} \) (if in parallel)
- Circuit D: \( R_{eq} = \frac{2R}{3} \)

6. **Determine the Minimum \( R_{eq} \):**
- The equivalent resistances are:
- Circuit A: \( \frac{R}{2} \)
- Circuit B: \( 2R \)
- Circuit C: \( \frac{R}{2} \)
- Circuit D: \( \frac{2R}{3} \)

- The minimum equivalent resistance is \( \frac{R}{2} \) from Circuit A and Circuit C.

7. **Conclusion:**
- Since both Circuit A and Circuit C have the same minimum equivalent resistance, they will dissipate the greatest power. However, if we are to choose one based on the problem statement, we can conclude that Circuit A is the answer.

### Final Answer:
**Circuit A dissipates the greatest power.**