If `C` and `L` denote capacitance and inductance respectively, then the dimensions of `LC` are

Options

A. `[M^(0)L^(0)T^(0)]`
B. `[M^(0)L^(2)T^(-2)]`
C. `[MLT^(-2)]`
D. `[M^(0)L^(0)T^(2)]`

Explanation

To find the dimensions of the product of capacitance \( C \) and inductance \( L \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Capacitance \( C \) is defined as the ability of a system to store charge per unit voltage. Its dimension can be derived from the formula:
\[ C = \frac{Q}{V} \]
where \( Q \) is charge and \( V \) is voltage.

2. **Determine the Dimensions of Charge and Voltage**:
- The dimension of charge \( Q \) is given by:
\[ [Q] = [I][T] = A \cdot s \]
where \( I \) is current (in amperes) and \( T \) is time (in seconds).
- The dimension of voltage \( V \) can be expressed as:
\[ V = \frac{W}{Q} = \frac{[M][L^2][T^{-2}]}{[Q]} = \frac{[M][L^2][T^{-2}]}{[I][T]} = [M][L^2][T^{-3}][I^{-1}] \]

3. **Calculate the Dimension of Capacitance \( C \)**:
- Substituting the dimensions of \( Q \) and \( V \) into the capacitance formula:
\[ [C] = \frac{[Q]}{[V]} = \frac{[I][T]}{[M][L^2][T^{-3}][I^{-1}]} = \frac{[I^2][T^4]}{[M][L^2]} \]

4. **Understand the Definition of Inductance**:
- Inductance \( L \) is defined as the ratio of the induced electromotive force to the rate of change of current. Its dimension can be derived from:
\[ L = \frac{V}{\frac{dI}{dt}} \]
- Thus, the dimension of inductance can be expressed as:
\[ [L] = \frac{[V]}{[I][T^{-1}]} = \frac{[M][L^2][T^{-3}][I^{-1}]}{[I][T^{-1}]} = \frac{[M][L^2][T^{-2}]}{[I^2]} \]

5. **Calculate the Dimensions of the Product \( LC \)**:
- Now we can find the dimensions of the product \( LC \):
\[ [LC] = [L] \cdot [C] = \left(\frac{[M][L^2][T^{-2}]}{[I^2]}\right) \cdot \left(\frac{[I^2][T^4]}{[M][L^2]}\right) \]
- Simplifying this expression:
\[ [LC] = \frac{[M][L^2][T^{-2}][I^2][T^4]}{[I^2][M][L^2]} = [T^2] \]

6. **Final Result**:
- The dimensions of \( LC \) are:
\[ [LC] = [T^2] \]

### Conclusion:
The dimensions of the product of capacitance \( C \) and inductance \( L \) are \( [T^2] \).