Animated Solution for Physics - Current Electricity: If a wire is stretched to make it 0.1% longer, its resistance will

Imagine you are holding a piece of chewing gum. When you pull it from both ends, it gets longer, but it also gets noticeably thinner. The exact same physical reality applies to a conducting wire. When a wire is stretched, its length $l$ increases, and its cross-sectional area $A$ decreases.

However, there is a fundamental constraint governing this deformation: the total amount of material cannot change. This means the volume of the wire remains absolutely constant. Mathematically, we express this as $V=A\cdot l=\text{constant}$. This single realization is the key to unlocking the entire problem.

We know the standard formula for the resistance of a wire is given by:

Where $\rho$ is the resistivity of the material. If we try to use this formula directly, we run into a problem: both $l$ and $A$ are changing simultaneously. Tracking two variables is cumbersome.

To elegantly bypass this, we can manipulate the equation to introduce our constant volume $V$. By multiplying both the numerator and the denominator by the length $l$, we get:

Since $A\cdot l$ is simply the volume $V$, the equation transforms into:

Now, look at the beauty of this new equation. The resistivity $\rho$ is a constant property of the material, and the volume $V$ is constant because the wire is merely being stretched. Therefore, we can confidently state that the resistance is directly proportional to the square of the length:

The problem states that the wire is stretched to make it $0.1\%$ longer. This is a microscopic change. Whenever we deal with percentage changes that are very small (typically less than $5\%$), we can deploy the powerful tools of error analysis and differential calculus.

Taking the natural logarithm of $R=\text{constant}\cdot l^2$ and differentiating, we get the fractional change:

Multiplying by $100$ gives us the percentage change:

This tells us that the percentage change in resistance is exactly twice the percentage change in length.

We are given that the percentage increase in length is $0.1\%$. Let's substitute this directly into our approximation:

The positive sign indicates an increase. Therefore, the resistance of the wire will increase by $0.2\%$.

It is crucial to understand the limitations of the tools we use. The $2\times$ multiplier is a brilliant shortcut, but it is an approximation. If the question had stated that the wire was stretched by $20\%$, using this shortcut would yield a $40\%$ increase, which is mathematically incorrect.

For large changes, you must use the exact ratio method:

If $l$ increases by $20\%$, then $l^{\prime }=1.2l$. The new resistance would be $R^{\prime }=(1.2{)}^{2}R=1.44R$, which is a $44\%$ increase, not $40\%$. Always check the magnitude of the change before choosing your mathematical weapon!