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(D) The magnetic field at the centre of a circular coil of radius $R$ carrying current $I$ is given by $B = \frac{\mu_0 I}{2 R}$.For the first coil wi

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$\frac{\sqrt{5} \mu_0 I}{2 R}$

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(D) The magnetic field at the centre of a circular coil of radius $R$ carrying current $I$ is given by $B = \frac{\mu_0 I}{2 R}$.For the first coil with current $I$,the magnetic field is $B_1 = \frac{\mu_0 I}{2 R}$.For the second coil with current $2I$,the magnetic field is $B_2 = \frac{\mu_0 (2I)}{2 R} = \frac{\mu_0 I}{R}$.Since the planes of the two coils are at right angles to each other,the magnetic fields $B_1$ and $B_2$ are perpendicular to each other.The resultant magnetic field $B$ at the centre is given by $B = \sqrt{B_1^2 + B_2^2}$.Substituting the values:$B = \sqrt{\left(\frac{\mu_0 I}{2 R}\right)^2 + \left(\frac{\mu_0 I}{R}\right)^2}$$B = \sqrt{\left(\frac{\mu_0 I}{2 R}\right)^2 + \left(\frac{2 \mu_0 I}{2 R}\right)^2}$$B = \frac{\mu_0 I}{2 R} \sqrt{1^2 + 2^2}$$B = \frac{\sqrt{5} \mu_0 I}{2 R}$.

Two similar coils each of radius $R$ are lying concentrically with their planes at right angles to each other. The currents flowing in them are $I$ and $2I$. The resultant magnetic field of induction at the centre will be ($\mu_0 =$ Permeability of vacuum).

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