Magnetic Properties of Matter

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 $4\%$.