The resistance R of a uniform wire of length l, cross-sectional area A, and resistivity ρ is given by :
\(R = \frac{\rho l}{A}\)
The resistance of a conductor depends on its material, its length and its thickness. Experiments show that resistance is directly proportional to the length of the wire and inversely proportional to its area of cross-section.
Step 1 — From the two proportionalities, \(R \propto \frac{l}{A}\).
Step 2 — The constant of proportionality is the resistivity \(\rho\) of the material, so \(R = \frac{\rho l}{A}\). Its SI unit follows as ohm metre.
The expression \(\frac{\rho A}{l}\) wrongly makes resistance grow with thickness and fall with length; \(\frac{\rho}{A}\) ignores length altogether, and \(\frac{A}{\rho}\) even makes resistance fall as resistivity rises.
\(R = \frac{\rho l}{A}\) option is correct.
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