A capacitor of capacitance C is fully charged by a 200 V battery. It is then discharged through a small coil of resistance wire embedded in a thermally insulated block of specific heat `2.5xx10^(2)Jkg^(-1)K^(-1)` and of mass 0.1 kg. if the temperature of the block rises by 0.4 K, what is the value of C?

Options

A. 500 F
B. 500 `mu`F
C. 50 F
D. 50 `mu`F

Answer

To solve the problem, we need to find the capacitance \( C \) of a capacitor that has been fully charged and then discharged through a resistance, causing a temperature rise in a thermally insulated block.

### Step-by-Step Solution:

1. **Understanding the Energy Stored in the Capacitor**:
The energy \( U \) stored in a capacitor is given by the formula:
\[ U = \frac{1}{2} C V^2 \]
where \( C \) is the capacitance and \( V \) is the voltage across the capacitor.

2. **Energy Converted to Heat**:
When the capacitor discharges through the resistance, this energy is converted into heat, which raises the temperature of the block. The heat energy \( Q \) absorbed by the block can be calculated using the formula:
\[ Q = m \cdot S \cdot \Delta T \]
where:
- \( m \) is the mass of the block,
- \( S \) is the specific heat capacity of the block,
- \( \Delta T \) is the change in temperature.

3. **Setting Up the Equation**:
Since the energy stored in the capacitor is equal to the heat absorbed by the block, we can set the two expressions equal:
\[ \frac{1}{2} C V^2 = m \cdot S \cdot \Delta T \]

4. **Substituting Known Values**:
We know:
- \( V = 200 \, \text{V} \)
- \( m = 0.1 \, \text{kg} \)
- \( S = 2.5 \times 10^2 \, \text{J/kg/K} \)
- \( \Delta T = 0.4 \, \text{K} \)

Plugging in these values:
\[ \frac{1}{2} C (200)^2 = 0.1 \cdot (2.5 \times 10^2) \cdot (0.4) \]

5. **Calculating the Right Side**:
First, calculate the right side:
\[ 0.1 \cdot 2.5 \times 10^2 \cdot 0.4 = 0.1 \cdot 250 \cdot 0.4 = 10 \]

6. **Calculating the Left Side**:
Now calculate the left side:
\[ \frac{1}{2} C (200)^2 = \frac{1}{2} C \cdot 40000 = 20000 C \]

7. **Setting the Equations Equal**:
Now we have:
\[ 20000 C = 10 \]

8. **Solving for \( C \)**:
To find \( C \), divide both sides by 20000:
\[ C = \frac{10}{20000} = 0.0005 \, \text{F} = 500 \, \mu\text{F} \]

### Final Answer:
The value of the capacitance \( C \) is \( 500 \, \mu\text{F} \).

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