The amount of work done in increasing the voltage across the plates of capacitor from 5 V to 10 V is W. The work done in increasing it from 10 V to 15 V will be

Options

A. W
B. 0.6W
C. 1.25 W
D. 1.67 W

Answer

To solve the problem of finding the work done in increasing the voltage across the plates of a capacitor from 10 V to 15 V, we can follow these steps:

### Step 1: Understand the formula for work done on a capacitor
The work done (W) in charging a capacitor can be expressed using the formula:
\[ W = \frac{1}{2} C (V_2^2 - V_1^2) \]
where \( C \) is the capacitance, \( V_1 \) is the initial voltage, and \( V_2 \) is the final voltage.

### Step 2: Calculate the work done from 5 V to 10 V
Using the formula, we first calculate the work done in increasing the voltage from 5 V to 10 V:
- Here, \( V_1 = 5 \, \text{V} \) and \( V_2 = 10 \, \text{V} \).
- Substitute into the formula:
\[ W = \frac{1}{2} C (10^2 - 5^2) \]
\[ W = \frac{1}{2} C (100 - 25) \]
\[ W = \frac{1}{2} C (75) \]
\[ W = \frac{75C}{2} \]

### Step 3: Calculate the work done from 10 V to 15 V
Now, we calculate the work done in increasing the voltage from 10 V to 15 V:
- Here, \( V_1 = 10 \, \text{V} \) and \( V_2 = 15 \, \text{V} \).
- Substitute into the formula:
\[ W' = \frac{1}{2} C (15^2 - 10^2) \]
\[ W' = \frac{1}{2} C (225 - 100) \]
\[ W' = \frac{1}{2} C (125) \]
\[ W' = \frac{125C}{2} \]

### Step 4: Relate the two work done values
We know from the first calculation that:
\[ W = \frac{75C}{2} \]
Now we can express \( W' \) in terms of \( W \):
\[ W' = \frac{125C}{2} \]
To find the ratio of \( W' \) to \( W \):
\[ \frac{W'}{W} = \frac{\frac{125C}{2}}{\frac{75C}{2}} \]
The \( \frac{C}{2} \) cancels out:
\[ \frac{W'}{W} = \frac{125}{75} = \frac{5}{3} \approx 1.67 \]

### Conclusion
Thus, the work done in increasing the voltage from 10 V to 15 V is:
\[ W' = 1.67 W \]

### Final Answer
The work done in increasing the voltage from 10 V to 15 V will be \( 1.67 W \).

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