The fractional change in the magnetic field intensity at a distance $'r'$ from centre on the axis of current carrying coil of radius $'a'$ to the magnetic field intensity at the centre of the same coil is : (Take $r << a )$
JEE MAIN 2021, Diffcult
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$B _{\text {axis }}=\frac{\mu_{0} i R ^{2}}{2\left( R ^{2}+ x ^{2}\right)^{3 / 2}}$
$B _{\text {centre }}=\frac{\mu_{0} i }{2 R }$
$\therefore B _{\text {centre }}=\frac{\mu_{0} i }{2 a }$
$\therefore B _{ axis }=\frac{\mu_{0} ia ^{2}}{2\left( a ^{2}+ r ^{2}\right)^{3 / 2}}$
$\therefore$ fractional change in magnetic field $=$
$\frac{\frac{\mu_{0} i }{2 a }-\frac{\mu_{0} ia ^{2}}{2\left( a ^{2}+ r ^{2}\right)^{3 / 2}}}{{\frac{\mu_{0} i }{2 a }}}=1-\frac{1}{\left[1+\left(\frac{ r ^{2}}{ a ^{2}}\right)\right]^{3 / 2}}$
$\approx 1-\left[1-\frac{3}{2} \frac{ r ^{2}}{ a ^{2}}\right]=\frac{3}{2} \frac{ r ^{2}}{ a ^{2}}$
Note : $\left(1+\frac{ r ^{2}}{ a ^{2}}\right)^{-3 / 2} \approx\left(1-\frac{3}{2} \frac{ r ^{2}}{ a ^{2}}\right)$
[True only if $r << a ]$
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