If C is capacitance and q is charge, then the dimension of `q^(2)//C` is same as that of

Options

A. work
B. angular momentum
C. force
D. momentum

Answer

To solve the problem of finding the dimension of \( \frac{q^2}{C} \), where \( C \) is capacitance and \( q \) is charge, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the given quantities**:
- \( q \) is the charge.
- \( C \) is the capacitance.

2. **Recall the formula for capacitance**:
- The capacitance \( C \) is defined as:
\[ C = \frac{q}{V} \]
where \( V \) is the voltage across the capacitor.

3. **Express voltage in terms of charge and capacitance**:
- Rearranging the formula gives:
\[ V = \frac{q}{C} \]

4. **Recall the formula for energy stored in a capacitor**:
- The energy \( U \) stored in a capacitor is given by:
\[ U = \frac{1}{2} C V^2 \]

5. **Substitute \( V \) in the energy formula**:
- Substitute \( V = \frac{q}{C} \) into the energy formula:
\[ U = \frac{1}{2} C \left(\frac{q}{C}\right)^2 \]

6. **Simplify the expression**:
- This simplifies to:
\[ U = \frac{1}{2} C \cdot \frac{q^2}{C^2} = \frac{1}{2} \frac{q^2}{C} \]

7. **Determine the dimension of energy**:
- The dimension of energy \( U \) is given by:
\[ [U] = [M L^2 T^{-2}] \]

8. **Relate the dimensions**:
- Since \( U = \frac{1}{2} \frac{q^2}{C} \), we can equate the dimensions:
\[ \left[\frac{q^2}{C}\right] = [U] = [M L^2 T^{-2}] \]

9. **Conclusion**:
- Therefore, the dimension of \( \frac{q^2}{C} \) is the same as that of energy, which is \( [M L^2 T^{-2}] \).

### Final Answer:
The dimension of \( \frac{q^2}{C} \) is the same as that of energy, which is \( [M L^2 T^{-2}] \).