Current Electricity: Common Mistakes and Fixes (9)

Question

A wire of resistance $R$ is stretched uniformly so that its length doubles. What is the new resistance? A surprising number of students answer $2R$. Why is that wrong, and what is the correct value?

Solution — Step by Step

The new resistance is $4R$.

Why This Works

Resistance scales as $L/A$. When the wire is stretched, both factors change: $L$ doubles and $A$ halves. The combined effect multiplies $R$ by $2 \times 2 = 4$. Students who only track the length get $2R$; those who only track the area get $R/2$. Both are wrong.

The key physical fact is conservation of volume — the metal isn’t created or destroyed. If length goes up by factor $n$, area goes down by factor $n$, and resistance goes up by factor $n^2$.

Alternative Method

Use $R \propto L^2/V$ where $V$ is volume. Since $V$ is constant, $R \propto L^2$. Doubling length means $4\times$ resistance. This shortcut is JEE-gold.

Common Mistake

Treating $A$ as constant. The fix is to write the volume-conservation step before substituting in the resistance formula. Once $A' = A/n$ is on paper, the right answer is automatic. JEE Main 2024 had a variant: stretched to triple length, find the new resistance. Answer: $9R$, not $3R$.