JEE Main 2025 Jan 28 Shift 2, Physics Q10: Units And Dimensions

The point $P$ lies on the equatorial (perpendicular bisector) line of the bar magnet at distance $d$ from its centre. The half-length of the magnet is $l$, and both $d$ and $l$ are quantities we measure with a length scale, so both carry the stated 1% uncertainty.

Step 1: Write the field formula.

Using the short-dipole approximation for the equatorial field:

$B=\frac{{\mu }_0}{4\pi }\cdot \frac{M}{d^3}$, where $M=m(2l)$ is the magnetic moment.

Since the pole strength $m$ is a property of the magnet (not something we measured with a scale), only $l$ and $d$ contribute uncertainty. Combining constants:

$B\propto \frac{l}{d^3}$.

Step 2: Propagate the errors.

For a quantity of the form $B\propto l^1{d}^{-3}$, take the log and differentiate. The percentage error rule gives:

$\frac{\Delta B}{B}=\frac{\Delta l}{l}+3\frac{\Delta d}{d}$.

Notice both contributions add (we take absolute values, since errors can go either way and we want the worst case).

Step 3: Plug in the numbers.

We are told $\frac{\Delta l}{l}=1\%$ and $\frac{\Delta d}{d}=1\%$, so

$\frac{\Delta B}{B}=1\%+3(1\%)=4\%$.