If C and R denote capacitance and resistance respectively, then the dimensional formula of CR is

Options

A. `[ML^(0)L^(0)T]`
B. `[M^(0)L^(0)T^(0)]`
C. `[M^(0)L^(0)T^(-1)]`
D. Not expressible in terms of `[MLT]`

Explanation

To find the dimensional formula of the product of capacitance (C) and resistance (R), we first need to determine the dimensional formulas for both capacitance and resistance.

### Step 1: Determine the dimensional formula for capacitance (C)

Capacitance (C) is defined as the charge (Q) per unit potential difference (V). Thus, we can express it as:

\[ C = \frac{Q}{V} \]

Where:
- Charge (Q) has the dimensional formula: \([I \cdot T]\) (where I is current and T is time).
- Potential difference (V) is defined as work done (W) per unit charge (Q). The dimensional formula for work done is \([M \cdot L^2 \cdot T^{-2}]\) (where M is mass, L is length, and T is time).

Thus, we can write:

\[ V = \frac{W}{Q} = \frac{[M \cdot L^2 \cdot T^{-2}]}{[I \cdot T]} = [M \cdot L^2 \cdot T^{-3} \cdot I^{-1}] \]

Now substituting back into the equation for capacitance:

\[ C = \frac{Q}{V} = \frac{[I \cdot T]}{[M \cdot L^2 \cdot T^{-3} \cdot I^{-1}]} \]

This simplifies to:

\[ C = [M^{-1} \cdot L^{-2} \cdot T^{4} \cdot I^{2}] \]

### Step 2: Determine the dimensional formula for resistance (R)

Resistance (R) is defined as the potential difference (V) per unit current (I):

\[ R = \frac{V}{I} \]

Using the dimensional formula for potential difference (V) derived earlier:

\[ R = \frac{[M \cdot L^2 \cdot T^{-3} \cdot I^{-1}]}{[I]} = [M \cdot L^2 \cdot T^{-3} \cdot I^{-2}] \]

### Step 3: Calculate the dimensional formula for the product CR

Now, we can find the dimensional formula for the product of capacitance and resistance:

\[ CR = C \cdot R = [M^{-1} \cdot L^{-2} \cdot T^{4} \cdot I^{2}] \cdot [M \cdot L^2 \cdot T^{-3} \cdot I^{-2}] \]

Multiplying these together:

\[ CR = [M^{-1} \cdot L^{-2} \cdot T^{4} \cdot I^{2} \cdot M \cdot L^2 \cdot T^{-3} \cdot I^{-2}] \]

This simplifies to:

\[ CR = [M^{0} \cdot L^{0} \cdot T^{1} \cdot I^{0}] = [T] \]

### Final Answer

The dimensional formula of \( CR \) is:

\[ [T] \]