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Arrange the following in ascending order of magnetic induction at a point on axis of circular loop of radius (r ) at a distance (x) from its centre ( for the same currents in the loops)
a) $r = R; x = \sqrt{3} R$ b) $r = 2 R; x = \sqrt{5} R$ c) $r = \frac{R}{2}; x = 2 \sqrt{2} R$

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a, c, b

a, b, c

b, c, a

c, a, b

answer is D.

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Detailed Solution

Magnetic field at a point x from the centre is

$B = \frac{μ_{0} \cdot i \cdot a^{2}}{2 {(a^{2} + X^{2})}^{3 / 2}}$

$B_{1} = \frac{μ_{0} \cdot i \cdot R^{2}}{2 {(R^{2} + 3 R^{2})}^{3 / 2}} = \frac{μ_{0} \cdot i}{16 R}$

$B_{2} = \frac{μ_{0} \cdot i \cdot 4 R^{2}}{2 {(4 R^{2} + 5 R^{2})}^{3 / 2}} = \frac{2 μ_{0} \cdot i}{27 R}$

$B_{3} = \frac{μ_{0} \cdot i \cdot {(\frac{R}{2})}^{2}}{2 {(R^{2} / 4 + 8 R^{2})}^{3 / 2}} = \frac{μ_{0} \cdot i}{33^{3 / 2} R}$

$B_{3} < B_{1} < B_{2}$

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