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

Options

A. `[M^(0)L^(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 will follow these steps:

### Step 1: Find the dimensional formula of capacitance (C)
Capacitance (C) is defined as the charge (Q) per unit voltage (V):
\[ C = \frac{Q}{V} \]

### Step 2: Find the dimensional formula of charge (Q)
Charge (Q) can be expressed in terms of current (I) and time (t):
\[ Q = I \cdot t \]
The dimensional formula for current (I) is:
\[ [I] = A \]
Thus, the dimensional formula for charge (Q) is:
\[ [Q] = [I][t] = A \cdot T \]

### Step 3: Find the dimensional formula of voltage (V)
Voltage (V) can be expressed as energy (work done) per unit charge. However, it can also be derived from Ohm's law:
\[ V = I \cdot R \]
Where R is resistance. The dimensional formula for voltage can also be expressed as:
\[ V = \frac{Q}{t} \]
Thus, the dimensional formula for voltage (V) is:
\[ [V] = \frac{[Q]}{[t]} = \frac{A \cdot T}{T} = A \]

### Step 4: Combine to find the dimensional formula for capacitance (C)
Now substituting the dimensional formulas into the formula for capacitance:
\[ [C] = \frac{[Q]}{[V]} = \frac{A \cdot T}{A} = T \]

### Step 5: Find the dimensional formula of resistance (R)
Resistance (R) is defined as voltage (V) per unit current (I):
\[ R = \frac{V}{I} \]

Using the dimensional formula for voltage (V) and current (I):
\[ [R] = \frac{[V]}{[I]} = \frac{A}{A} = T^2 \]

### Step 6: Find the dimensional formula for the product CR
Now we can find the dimensional formula for the product of capacitance and resistance:
\[ [CR] = [C] \cdot [R] = T \cdot T^2 = T^3 \]

### Final Result
The dimensional formula of \( CR \) is:
\[ [CR] = T^3 \]

### Conclusion
Thus, the answer to the question is that the dimensional formula of \( CR \) is \( T^3 \).

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