Ans. Resistance $R$ of a conductor depends on the length '$l$' and area of cross section '$A$' of the conductor.
$$R \propto l$$
and $$R \propto \frac{1}{A}$$
$$\therefore R \propto \frac{l}{A}$$
$$\therefore R = \rho \times \frac{l}{A} \quad (\rho \text{ is constant})$$
$$\rho \text{ (Resistivity)} = \frac{RA}{l}$$
Where $\rho$ (rho) is called resistivity of the conductor it is also called as specific resistance.
If we put $l = 1m$ and $A = 1m^2$ then $R = \rho$
Conclusion
Thus, resistivity of a conductor is defined as the resistance of a conductor of unit length and unit area of cross section. The SI unit of resistivity is ohm – metre ($\Omega - m$).
Resistivity is the characteristic property of material. It is different for different materials.
Explanation:
Step 1: Understand the factors affecting resistance.
Imagine water flowing through a pipe. A longer pipe creates more friction (resistance). Similarly, electrical resistance ($R$) is directly proportional to the length ($l$) of the wire. Mathematically, $R \propto l$.
A wider pipe allows more water to flow easily. Similarly, a larger cross-sectional area ($A$) of a wire offers more room for electrons to travel, resulting in less resistance. Thus, resistance is inversely proportional to the area. Mathematically, $R \propto \frac{1}{A}$.Step 2: Combine the proportionalities.
Combining the two relationships from Step 1 gives us a joint proportionality statement: $R \propto \frac{l}{A}$.Step 3: Introduce the constant of proportionality.
To convert the proportionality sign ($\propto$) into an equal sign ($=$), we must multiply by a constant. For a given material at a constant temperature, this intrinsic property is called Resistivity, denoted by the Greek letter $\rho$ (rho).
So, the formula becomes: $R = \rho \frac{l}{A}$.Step 4: Rearrange the formula to solve for Resistivity ($\rho$).
To isolate $\rho$ on one side of the equation, multiply both sides by $A$ and divide by $l$.
$\rho = \frac{R \cdot A}{l}$Step 5: Define Resistivity theoretically and find its SI unit.
If we take a hypothetical piece of material that is $1\text{ meter}$ long ($l = 1m$) with a cross-sectional area of $1\text{ square meter}$ ($A = 1m^2$), the equation simplifies to $\rho = \frac{R \cdot 1}{1}$, which means $\rho = R$. Therefore, resistivity is numerically equal to the resistance of a unit cube of that specific material.
To deduce the SI unit, substitute the standard units into the rearranged formula:
$R$ is measured in Ohms ($\Omega$)
$A$ is measured in square meters ($m^2$)
$l$ is measured in meters ($m$)
Unit of $\rho = \frac{\Omega \cdot m^2}{m} = \Omega \cdot m$ (Ohm-meter).