Animated Solution for Physics - Current Electricity: A copper wire is stretched to make it $0.1\%$ longer. What is the percentage change in its resistance ?

Imagine you are holding a piece of copper wire. It has a certain length, let's call it $l$, and a certain thickness, which we measure as its cross-sectional area, $A$. The resistance $R$ of this wire—how much it fights against the flow of electricity—is given by the classic formula:

Here, $\rho$ (rho) is the resistivity of copper, a fundamental property of the material itself that doesn't change unless we heat it up or cool it down.

Volume Conservation

Now, the problem asks us to stretch the wire so that it becomes $0.1\%$ longer. A common trap many students fall into is thinking, "Well, if the length increases by $0.1\%$, the resistance must also increase by $0.1\%$ because $R$ is directly proportional to $l$."

When you stretch a wire, you are not magically creating more copper out of thin air. The total amount of copper—its volume $V$—must remain absolutely constant.

Since the volume of a cylinder is the product of its cross-sectional area and its length ($V=A\cdot l$), stretching the wire to make it longer forces it to become thinner. The area $A$ must decrease!

To handle this mathematically, we can eliminate the pesky variable $A$ from our resistance equation. We know that $A=\frac{V}{l}$. Let's substitute this back into our resistance formula:

Look at this beautiful new equation! Since $\rho$ and $V$ are both constants, we have discovered a profound relationship: When a wire is stretched, its resistance is proportional to the square of its length ($R\propto l^2$).

We are told the length increases by a tiny amount: $0.1\%$. Because this change is so small (much less than $5\%$), we don't need to do complicated algebraic expansions. We can borrow a brilliant trick from error analysis and calculus.

If a quantity $y$ depends on $x$ such that $y\propto x^n$, then for very small changes, the percentage change in $y$ is simply $n$ times the percentage change in $x$:

Applying this to our resistance equation $R\propto l^2$, the power of $2$ comes down to the front:

We are given that the percentage change in length is $0.1\%$. Let's plug that in:

Because the length increased, the resistance must also increase. Therefore, stretching the wire by $0.1\%$ causes its resistance to increase by $0.2\%$.

This is a favorite concept for JEE because it elegantly combines basic electricity with the geometry of volume conservation and the mathematical elegance of small approximations. Never forget: when you stretch a wire, it fights back twice as hard!