The physical quantity having the dimensions `[M^(–1)L^(–3)T^(3)A^(2)]` is

Options

A. resistance
B. resistivity
C. electrical conductivity
D. electromotive force

Answer

To solve the problem of identifying the physical quantity that has the dimensions \([M^{-1}L^{-3}T^{3}A^{2}]\), we will analyze the dimensions systematically.

### Step-by-Step Solution:

1. **Understanding the Dimensions**:
- The given dimensions are \([M^{-1}L^{-3}T^{3}A^{2}]\).
- Here, \(M\) represents mass, \(L\) represents length, \(T\) represents time, and \(A\) represents electric current (Ampere).

2. **Identifying Possible Physical Quantities**:
- We will consider common physical quantities that involve these dimensions. The most relevant quantities in the context of electrical properties are resistance, resistivity, electrical conductivity, and electromotive force (EMF).

3. **Calculating the Dimension of Resistance**:
- The dimension of resistance \(R\) is given by:
\[ R = \frac{V}{I} \]
where \(V\) (voltage) has dimensions of \([ML^{2}T^{-3}A^{-1}]\) and \(I\) (current) has dimensions of \([A]\).
- Therefore, the dimensions of resistance are:
\[ [R] = \frac{[ML^{2}T^{-3}A^{-1}]}{[A]} = [ML^{2}T^{-3}A^{-2}] \]

4. **Calculating the Dimension of Resistivity**:
- Resistivity \(\rho\) is defined as:
\[ \rho = R \cdot \frac{A}{L} \]
where \(A\) is the cross-sectional area \([L^{2}]\) and \(L\) is the length \([L]\).
- Thus, the dimensions of resistivity are:
\[ [\rho] = [R] \cdot \frac{[L^{2}]}{[L]} = [ML^{2}T^{-3}A^{-2}] \cdot [L] = [ML^{3}T^{-3}A^{-2}] \]

5. **Calculating the Dimension of Electrical Conductivity**:
- Electrical conductivity \(\sigma\) is the reciprocal of resistivity:
\[ \sigma = \frac{1}{\rho} \]
- Therefore, the dimensions of conductivity are:
\[ [\sigma] = \frac{1}{[ML^{3}T^{-3}A^{-2}]} = [M^{-1}L^{-3}T^{3}A^{2}] \]
- This matches the given dimensions.

6. **Calculating the Dimension of EMF**:
- EMF is defined as:
\[ EMF = \frac{W}{Q} \]
where \(W\) (work) has dimensions of \([ML^{2}T^{-2}]\) and \(Q\) (charge) has dimensions of \([IT] = [AT]\).
- Thus, the dimensions of EMF are:
\[ [EMF] = \frac{[ML^{2}T^{-2}]}{[AT]} = [ML^{2}T^{-3}A^{-1}] \]
- This does not match the given dimensions.

### Conclusion:
The physical quantity that has the dimensions \([M^{-1}L^{-3}T^{3}A^{2}]\) is **Electrical Conductivity**.