The resistance of a wire of length l , radius r , and resistivity $$\rho$$ is,

Correct option is D

The formula for the resistance $R$ of a wire is derived from the equation:
​$$R = \rho \frac{l}{A}$$​​

Where:

• $\rho$: Resistivity of the material of the wire.
• $l$: Length of the wire.
• $A$: Cross-sectional area of the wire.
• $$A = \pi r^2$$

The cross-sectional area of the wire, $A$, is given by:

Substituting $A$ into the resistance formula:
$$R = ρ\frac{ l}{\pi r^2}$$​

Thus, the correct option is D.

Resistance and Its Dependence

• Length (l): Resistance is directly proportional to the length of the wire. Longer wires have higher resistance.
• Cross-sectional Area (A): Resistance is inversely proportional to the area. Larger areas result in lower resistance.
• Resistivity (ρ): Resistance depends on the material and its resistivity, which is a constant for a specific material.

Role of Radius

• Since A=πr2A = \pi r^2$$A = \pi r^2$$​, the radius of the wire significantly affects the resistance. Smaller radii result in larger resistance values.

Applications of Resistance Formula

• Design of Electrical Circuits: Used in selecting wire dimensions to control current flow.
• Material Science: Helps in understanding the electrical properties of different materials.