In the diagrams, all the light bulbs are identical, and all cells are ideal and identical. In which circuit (a,b, c,d) will the bulbs be dimmest ?

Options

A.
B.
C.
D.

Explanation

To determine which circuit (A, B, C, or D) has the dimmest bulbs, we will analyze the power delivered to the bulbs in each circuit. The power (P) of a bulb can be expressed as:

\[ P = \frac{V^2}{R} \]

where \( V \) is the voltage across the bulb and \( R \) is the resistance of the bulb. Since all bulbs are identical, we can focus on the voltage across each bulb in each circuit.

### Step 1: Analyze Circuit A
- In Circuit A, there are 2 cells connected in series providing a total voltage of \( 2\epsilon \).
- There are 2 bulbs connected in series, so the voltage across each bulb is \( \frac{2\epsilon}{2} = \epsilon \).
- The power for each bulb is:
\[ P_A = \frac{\epsilon^2}{R} \]

### Step 2: Analyze Circuit B
- In Circuit B, there are 3 cells connected in series providing a total voltage of \( 2\epsilon \).
- There are 3 bulbs connected in series, so the voltage across each bulb is \( \frac{2\epsilon}{3} \).
- The power for each bulb is:
\[ P_B = \frac{\left(\frac{2\epsilon}{3}\right)^2}{R} = \frac{4\epsilon^2}{9R} \]

### Step 3: Analyze Circuit C
- In Circuit C, there are 3 cells connected in parallel providing a total voltage of \( \epsilon \).
- The voltage across each bulb is \( \frac{\epsilon}{3} \) (since they are in parallel).
- The power for each bulb is:
\[ P_C = \frac{\left(\frac{\epsilon}{3}\right)^2}{R} = \frac{\epsilon^2}{9R} \]

### Step 4: Analyze Circuit D
- In Circuit D, there are 2 cells connected in parallel providing a total voltage of \( 2\epsilon \).
- The voltage across each bulb is \( 2\epsilon \).
- The power for each bulb is:
\[ P_D = \frac{(2\epsilon)^2}{R} = \frac{4\epsilon^2}{R} \]

### Step 5: Compare the Powers
Now we compare the powers calculated for each circuit:
- \( P_A = \frac{\epsilon^2}{R} \)
- \( P_B = \frac{4\epsilon^2}{9R} \)
- \( P_C = \frac{\epsilon^2}{9R} \)
- \( P_D = \frac{4\epsilon^2}{R} \)

From the calculations:
- \( P_A > P_B > P_C < P_D \)

### Conclusion
The circuit with the dimmest bulbs is Circuit C, as it has the least power delivered to each bulb.

### Final Answer
The bulbs will be dimmest in Circuit C.

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