Dimension of `CR` are those of

Options

A. Frequency
B. Energy
C. Time period
D. Current

Answer

To find the dimensions of capacitance \( C_R \), we start with the fundamental relationship that defines capacitance:

1. **Understanding Capacitance**:
The capacitance \( C \) is defined as the charge \( Q \) stored per unit potential \( V \):
\[ C = \frac{Q}{V} \]

2. **Dimensions of Charge**:
The dimension of charge \( Q \) can be derived from the relationship of current \( I \) and time \( T \):
\[ Q = I \cdot T \]
The dimension of current \( I \) is represented as \( [I] \) and time \( T \) is represented as \( [T] \). Therefore, the dimension of charge \( Q \) is:
\[ [Q] = [I][T] \]

3. **Dimensions of Potential**:
The potential \( V \) can be defined in terms of resistance \( R \) and current \( I \):
\[ V = I \cdot R \]
The dimension of resistance \( R \) can be expressed as:
\[ R = \frac{V}{I} \]
Rearranging gives:
\[ V = R \cdot I \]
The dimension of resistance \( R \) can be derived from Ohm's law, but for our purpose, we can express potential as:
\[ [V] = [I][R] \]

4. **Substituting Dimensions into Capacitance**:
Now substituting the dimensions of \( Q \) and \( V \) into the capacitance formula:
\[ C = \frac{Q}{V} = \frac{[I][T]}{[I][R]} \]
Here, the \( [I] \) cancels out:
\[ C = \frac{[T]}{[R]} \]

5. **Final Dimensions of Capacitance**:
The dimension of capacitance \( C \) can be simplified to:
\[ [C] = [T][R]^{-1} \]
Since the dimension of resistance \( R \) is \( [M][L^2][T^{-3}][I^{-2}] \), we can express the dimension of capacitance as:
\[ [C] = [M^{-1}][L^{-2}][T^4][I^2] \]

Thus, the dimensions of \( C_R \) are those of capacitance, which can be expressed as:
\[ [C_R] = [M^{-1}][L^{-2}][T^4][I^2] \]