JEE Main 2025 — Magnetic Properties Of Matter Question with Solution

To determine the uncertainty of the magnetic field at point P, we consider two approaches:

Without considering uncertainty in length $\ell$:

The magnetic field $B$ at point P is given by:

$B=\frac{{\mu }_0}{4\pi }\frac{m}{r^3}$

Where $B$ is inversely proportional to $r^3$:

$B\propto \frac{1}{r^3}$

The relative uncertainty in $B$, expressed as a percentage, is calculated by:

$\frac{\Delta B}{B}=3\times \left(\frac{\Delta r}{r}\right)$

Given that the relative uncertainty in length measurement is $1\%$, the uncertainty in $B$ becomes:

$\%\text{ uncertainty in }B=3\%$

Method-2: Considering uncertainty in length $\ell$:

Here, we also account for the uncertainty in $\ell$:

$B\propto \frac{1}{r^3}$

Thus, the combined relative uncertainty, incorporating $\ell$, is:

$\frac{\Delta B}{B}=\frac{\Delta \ell }{\ell }+3\times \left(\frac{\Delta r}{r}\right)$

Substituting the given uncertainties:

$\frac{\Delta B}{B}=1+3\times 1=4\%$

Therefore, the percentage uncertainty in $B$, considering the length $\ell$, is $4\%$.