The dimensional formula for resistivity of conductor is

Options

A. `[M L^(2) T^(-2) A^(-2)]`
B. `[M L^(3) T^(-3) A^(-2)]`
C. `[M L^(-2) T^(-2) A^(2)]`
D. `[M L^(2) T^(-2) A^(-3)]`

Explanation

To find the dimensional formula for the resistivity of a conductor, we can follow these steps:

### Step 1: Understand the formula for resistivity
Resistivity (ρ) is defined as:
\[ \rho = R \frac{A}{L} \]
where:
- \( R \) is the resistance,
- \( A \) is the cross-sectional area of the conductor,
- \( L \) is the length of the conductor.

### Step 2: Find the dimensional formula for resistance (R)
Resistance \( R \) is defined as:
\[ R = \frac{V}{I} \]
where:
- \( V \) is the voltage,
- \( I \) is the current.

The dimensional formula for voltage \( V \) can be expressed as:
\[ V = E \cdot d \]
where \( E \) is the electric field and \( d \) is the distance. The electric field \( E \) has dimensions of \( \frac{ML}{T^3A} \) (derived from \( E = \frac{F}{q} \) and \( F = ma \)).

Thus, the dimensional formula for voltage becomes:
\[ [V] = [E][d] = \left[\frac{ML}{T^3A}\right][L] = \frac{ML^2}{T^3A} \]

Now, for current \( I \), the dimensional formula is:
\[ [I] = A \]

So, the dimensional formula for resistance \( R \) is:
\[ [R] = \frac{[V]}{[I]} = \frac{\frac{ML^2}{T^3A}}{A} = \frac{ML^2}{T^3A^2} \]

### Step 3: Substitute the dimensional formula of resistance into the resistivity formula
Now substituting \( R \) into the resistivity formula:
\[ [\rho] = [R] \cdot \frac{[A]}{[L]} \]
We know:
- The dimensional formula for area \( A \) is \( [A] = L^2 \).
- The dimensional formula for length \( L \) is \( [L] = L \).

Thus:
\[ [\rho] = \left[\frac{ML^2}{T^3A^2}\right] \cdot \frac{L^2}{L} = \frac{ML^2}{T^3A^2} \cdot L = \frac{ML^3}{T^3A^2} \]

### Final Result
The dimensional formula for resistivity \( \rho \) is:
\[ [\rho] = M L^3 T^{-3} A^{-2} \]