A bar magnet has total length $2 l=20$ units and the field point $P$ is at a distance $d=10$ 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 :
Select the correct option:
A
$4 \%$
B
$5 \%$
C
$10 \%$
D
$3 \%$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
Method-1 :
Without considering uncertainty in $\ell$.
$\mathrm{B}=\frac{\mu_{0}}{4 \pi} \frac{\mathrm{~m}}{\mathrm{r}^{3}}$
$\mathrm{B} \propto \frac{1}{\mathrm{r}^{3}}$
$\frac{\Delta \mathrm{B}}{\mathrm{B}}=3 \times\left(\frac{\Delta \mathrm{r}}{\mathrm{r}}\right)$
% uncertainty in B = 3% Method-2 :
With considering uncertainty in $\ell$.
$\mathrm{B} \propto \frac{1}{\mathrm{r}^{3}}$
$\frac{\Delta \mathrm{B}}{\mathrm{B}}=\frac{\Delta \ell}{\ell}+3 \times\left(\frac{\Delta \mathrm{r}}{\mathrm{r}}\right)=1+3 \times 1=4 \%$
$\%$ uncertainty in $B=4 \%$
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A bar magnet has total length $2 l=20$ units and the field point $P$ is at a distance $d=10$ 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 :
