In the following `'I'` refers to current and other symbols have their usual meaning. Chosse the option that corresponds to the dimensions of electrical conductivity:

Options

A. `M^(-1)L^(-3)T^(3)I^(2)`
B. `M^(-1)L^(3)T^(3)I`
C. `ML^(-3)T^(-3)I^(2)`
D. `M^(-1)L^(-3)T^(3)I`

Explanation

To find the dimensions of electrical conductivity, we start from the relationship between resistance, resistivity, and conductivity. The electrical conductivity (σ) is defined as the inverse of resistivity (ρ), and resistivity is related to resistance (R) through the formula:

\[ \sigma = \frac{1}{\rho} \]

Resistance (R) can be expressed in terms of resistivity (ρ) as follows:

\[ R = \frac{\rho L}{A} \]

Where:
- \(R\) = Resistance
- \(\rho\) = Resistivity
- \(L\) = Length of the conductor
- \(A\) = Cross-sectional area of the conductor

From this, we can express resistivity in terms of resistance:

\[ \rho = \frac{R A}{L} \]

Now, substituting this into the equation for conductivity:

\[ \sigma = \frac{1}{\rho} = \frac{A}{R L} \]

Next, we need to find the dimensions of each term in this equation.

1. **Dimensions of Resistance (R)**:
The formula for power in terms of current (I) and resistance (R) is given by:

\[ P = I^2 R \]

Power (P) has dimensions of work done per unit time:

\[ [P] = \frac{[W]}{[T]} = \frac{ML^2T^{-2}}{T} = ML^2T^{-3} \]

Rearranging the power equation gives us:

\[ R = \frac{P}{I^2} \]

Thus, the dimensions of resistance can be expressed as:

\[ [R] = \frac{[P]}{[I]^2} = \frac{ML^2T^{-3}}{I^2} \]

2. **Substituting for Resistance**:
Now substituting the dimensions of resistance into the conductivity equation:

\[ [\sigma] = \frac{[A]}{[R][L]} = \frac{[A]}{\left(\frac{ML^2T^{-3}}{I^2}\right)[L]} \]

The dimensions of area (A) are:

\[ [A] = L^2 \]

Thus, substituting this into the conductivity equation gives:

\[ [\sigma] = \frac{L^2}{\left(\frac{ML^2T^{-3}}{I^2}\right)L} = \frac{L^2 I^2 T^{-3}}{ML^3} \]

3. **Simplifying the Dimensions**:
Simplifying this expression:

\[ [\sigma] = \frac{I^2}{ML} \]

4. **Final Dimensions of Electrical Conductivity**:
Therefore, the dimensions of electrical conductivity are:

\[ [\sigma] = M^{-1} L^{-3} T^{3} I^{2} \]

### Final Answer:
The dimensions of electrical conductivity are:

\[ M^{-1} L^{-3} T^{3} I^{2} \]