The dimensional formula fo resistivity in terms of `M,L,T` and `Q` where `Q` stands for the dimensions of charge is

Options

A. `[ML^(3)T^(-1)Q^(-2)]`
B. `[ML^(2)T^(-2)Q^(2)]`
C. `[MLT^(-2)Q^(-2)]`
D. `[ML^(2)T^(-2)Q^(-1)]`

Explanation

To find the dimensional formula for resistivity in terms of mass (M), length (L), time (T), and charge (Q), we will follow these steps:

### Step 1: Understand the relationship between resistivity, resistance, area, and length.
The formula for resistance (R) is given by:
\[ R = \rho \frac{L}{A} \]
Where:
- \( \rho \) is the resistivity,
- \( L \) is the length,
- \( A \) is the area of cross-section.

From this, we can rearrange the formula to express resistivity:
\[ \rho = R \frac{A}{L} \]

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

### Step 3: Find the dimensions of voltage (V).
Voltage is defined as work done (W) per unit charge (Q):
\[ V = \frac{W}{Q} \]
The dimensions of work done (W) are given by:
\[ W = \text{Force} \times \text{Distance} = (MLT^{-2}) \times L = ML^2T^{-2} \]
Thus, the dimensions of voltage become:
\[ V = \frac{ML^2T^{-2}}{Q} \]
So, the dimensions of voltage are:
\[ [V] = ML^2T^{-2}Q^{-1} \]

### Step 4: Find the dimensions of current (I).
Current is defined as charge (Q) per unit time (T):
\[ I = \frac{Q}{T} \]
Thus, the dimensions of current are:
\[ [I] = QT^{-1} \]

### Step 5: Substitute the dimensions of voltage and current into the resistance formula.
Now, substituting the dimensions of voltage and current into the resistance formula:
\[ R = \frac{[V]}{[I]} = \frac{ML^2T^{-2}Q^{-1}}{QT^{-1}} \]
This simplifies to:
\[ [R] = \frac{ML^2T^{-2}Q^{-1}}{QT^{-1}} = ML^2T^{-1}Q^{-2} \]

### Step 6: Substitute the dimensions of resistance, area, and length into the resistivity formula.
Now we have:
- Dimensions of resistance \( [R] = ML^2T^{-1}Q^{-2} \)
- Dimensions of area \( [A] = L^2 \)
- Dimensions of length \( [L] = L \)

Substituting these into the resistivity formula:
\[ \rho = R \frac{A}{L} = \left(ML^2T^{-1}Q^{-2}\right) \cdot \frac{L^2}{L} \]
This simplifies to:
\[ \rho = ML^2T^{-1}Q^{-2} \cdot L = ML^3T^{-1}Q^{-2} \]

### Final Result
Thus, the dimensional formula for resistivity \( \rho \) is:
\[ [\rho] = ML^3T^{-1}Q^{-2} \]