Given that `C` denotes capacitance of a capacitor and `V` is the potential difference across its plates. Then the dimensions of `CV^(2)` are same as that of

Options

A. momentum
B. power
C. forces
D. torque

Answer

To find the dimensions of \( CV^2 \), where \( C \) is the capacitance and \( V \) is the potential difference, we can follow these steps:

### Step-by-Step Solution

1. **Identify the dimensions of capacitance \( C \)**:
The capacitance \( C \) is defined as the charge \( Q \) per unit potential difference \( V \). The formula is:
\[ C = \frac{Q}{V} \]
The dimension of charge \( Q \) is \( [I][T] \) (where \( I \) is current and \( T \) is time). The dimension of potential difference \( V \) is \( [M][L^2][T^{-3}][I^{-1}] \) (derived from the formula \( V = W/Q \), where \( W \) is work done, and its dimension is \( [M][L^2][T^{-2}] \)).

Therefore, the dimension of capacitance \( C \) can be expressed as:
\[ [C] = \frac{[Q]}{[V]} = \frac{[I][T]}{[M][L^2][T^{-3}][I^{-1}]} = [M^{-1}][L^{-2}][T^4][I^2] \]

2. **Identify the dimensions of potential difference \( V \)**:
As mentioned earlier, the dimension of potential difference \( V \) is:
\[ [V] = [M][L^2][T^{-3}][I^{-1}] \]

3. **Calculate the dimensions of \( V^2 \)**:
To find \( V^2 \), we square the dimensions of \( V \):
\[ [V^2] = ([M][L^2][T^{-3}][I^{-1}])^2 = [M^2][L^4][T^{-6}][I^{-2}] \]

4. **Combine the dimensions of \( C \) and \( V^2 \)**:
Now we can find the dimensions of \( CV^2 \):
\[ [CV^2] = [C] \cdot [V^2] = [M^{-1}][L^{-2}][T^4][I^2] \cdot [M^2][L^4][T^{-6}][I^{-2}] \]

5. **Multiply the dimensions**:
When we multiply the dimensions, we combine the powers of each base unit:
\[ [CV^2] = [M^{-1} \cdot M^2][L^{-2} \cdot L^4][T^4 \cdot T^{-6}][I^2 \cdot I^{-2}] \]
This simplifies to:
\[ [CV^2] = [M^{1}][L^{2}][T^{-2}][I^{0}] = [M][L^{2}][T^{-2}] \]

6. **Conclusion**:
The dimensions of \( CV^2 \) are the same as that of energy, which is given by:
\[ [E] = [M][L^2][T^{-2}] \]

### Final Answer:
The dimensions of \( CV^2 \) are the same as that of energy.