Question 147: Magnetism and Matter

Question

A bar magnet has total length $2 l=20$ units and the field point P is at a distance $\mathrm{d}=10$ units from the centre of the magnet. If the relative uncertainty of length measurement is $1 \%$, then uncertainty of the magnetic field at point P is :

Accepted Answer

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 	imes \left(\frac{\Delta r}{r}ight) $
Given that the relative uncertainty in length measurement is $1\%$, the uncertainty in $ B $ becomes:
$ \% 	ext{ 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 	imes \left(\frac{\Delta r}{r}ight) $
Substituting the given uncertainties:
$ \frac{\Delta B}{B} = 1 + 3 	imes 1 = 4\% $
Therefore, the percentage uncertainty in $ B $, considering the length $\ell$, is $4\%$.

Answer

10%

Answer

5%

Answer

3%

Answer

4%

Magnetism and Matter

A bar magnet has total length 2 l=20 units and the field point P is at a distance \mathrm{d}=10 units from the centre of the magnet. If the relative uncertainty of length measurement is 1 \%, then uncertainty of the magnetic field at point P is :

Choose the correct answer: