If c and R denote capacity and resistance the dimensions of CR are :

Options

A. `[M^(0)L^(0)T^(-1)]`
B. `[M^(1)L^(0)T^(2)]`
C. `[M^(0)L^(0)T^(1)]`
D. Not expressable in terms of `[M], [L]` and `[T]`

Explanation

To find the dimensions of the product of capacity (C) and resistance (R), we will follow these steps:

### Step 1: Determine the dimension of Capacity (C)
Capacity (C) is defined as the charge (Q) per unit voltage (V). Mathematically, it can be expressed as:
\[ C = \frac{Q}{V} \]

The dimension of charge (Q) is given by the dimension of current (I) multiplied by time (T):
\[ [Q] = [I][T] = [A][T] \]

Voltage (V) can be expressed in terms of work (W) and charge (Q):
\[ V = \frac{W}{Q} \]
Where the dimension of work (W) is energy, which has the dimension:
\[ [W] = [M][L^2][T^{-2}] \]

Substituting the dimension of charge into the equation for voltage:
\[ [V] = \frac{[M][L^2][T^{-2}]}{[A][T]} = \frac{[M][L^2][T^{-2}]}{[A][T]} = [M][L^2][T^{-3}][A^{-1}] \]

Now substituting the dimensions of Q and V back into the equation for C:
\[ [C] = \frac{[A][T]}{[M][L^2][T^{-3}][A^{-1}]} = \frac{[A^2][T^4]}{[M][L^2]} \]
Thus, the dimension of capacity is:
\[ [C] = [M^{-1}][L^{-2}][T^4][A^2] \]

### Step 2: Determine the dimension of Resistance (R)
Resistance (R) is defined as the ratio of voltage (V) to current (I):
\[ R = \frac{V}{I} \]

Substituting the dimension of voltage:
\[ [R] = \frac{[M][L^2][T^{-3}][A^{-1}]}{[A]} = [M][L^2][T^{-3}][A^{-2}] \]

### Step 3: Calculate the dimensions of the product CR
Now we can find the dimensions of the product \( CR \):
\[ [CR] = [C] \times [R] \]
Substituting the dimensions we found:
\[ [CR] = \left([M^{-1}][L^{-2}][T^4][A^2]\right) \times \left([M][L^2][T^{-3}][A^{-2}]\right) \]

### Step 4: Simplify the dimensions
Now, we will simplify the expression:
\[ [CR] = [M^{-1}][M][L^{-2}][L^2][T^4][T^{-3}][A^2][A^{-2}] \]
Combining like terms:
- For mass (M): \( M^{-1} \times M = M^{0} \)
- For length (L): \( L^{-2} \times L^2 = L^{0} \)
- For time (T): \( T^4 \times T^{-3} = T^{1} \)
- For current (A): \( A^2 \times A^{-2} = A^{0} \)

Thus, we have:
\[ [CR] = [M^0][L^0][T^1][A^0] = [T] \]

### Final Answer
The dimensions of the product of capacity and resistance (CR) are:
\[ [CR] = [T] \]