The dimensions of resistivity in terms of M, L, T, and Q, where Q stands for the dimensions of charge is A. M L^3 T^1 Q^2 B. M L^3 T^2 Q^1 C. M L^2 T^1 Q^1 D. M L T^1 Q^1.
Question
The dimensions of resistivity in terms of $M, L, T$ and $Q$, where $Q$ stands for the dimensions of charge is
A. $M L^{3} T^{1} Q^{2}$ B. $M L^{3} T^{2} Q^{1}$
C. $M L^{2} T^{1} Q^{1}$ D. $M L T^{1} Q^{1}$.
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Answer
To determine the dimensions of resistivity ($\rho$) in terms of $M$, $L$, $T$, and $Q$ (where $Q$ represents the dimensions of charge), let's first understand the relationship among the physical quantities:
Resistivity ($\rho$) can be expressed as: $$ \rho = \frac{R \cdot A}{L} $$ where:
 $R$ is the resistance,
 $A$ is the crosssectional area,
 $L$ is the length.
We need the dimensions for each of these terms.

Dimensions of Resistance ($R$)
 Ohm's law states that $R = \frac{V}{I}$, where $V$ is voltage and $I$ is current.
 Voltage ($V$) has dimensions of $\left[ M L^2 T^{3} Q^{1} \right]$.
 Current ($I$) has dimensions of $\left[ Q T^{1} \right]$.
So, the dimensions of resistance $R$ are: $$ \left[ R \right] = \frac{[V]}{[I]} = \frac{\left[ M L^2 T^{3} Q^{1} \right]}{\left[ Q T^{1} \right]} = M L^2 T^{3} Q^{2} $$

Dimensions of Area ($A$)
 Area is length squared, so: $$ \left[ A \right] = L^2 $$

Dimensions of Length ($L$)
 Length has dimensions $[L]$.
Combining these, the dimensions of resistivity $\rho$ are: $$ \rho = \frac{R \cdot A}{L} = \frac{[R] \cdot [A]}{[L]} $$ Substituting the dimensions obtained: $$ \left[ \rho \right] = \frac{\left[M L^2 T^{3} Q^{2}\right] \cdot \left[L^2\right]}{[L]} = \frac{M L^4 T^{3} Q^{2}}{L} = M L^3 T^{3} Q^{2} $$
Thus, the correct dimensional formula for resistivity is: $$ M L^3 T^{3} Q^{2} $$
Hence, the correct option is:
A. $M L^{3} T^{1} Q^{2}$
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