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Two Identical Coils P and Q Each of Radius R Are Lying in Perpendicular Planes Such that They Have a Common Centre.

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Question

Two identical coils P and Q each of radius R are lying in perpendicular planes such that they have a common centre. Find the magnitude and direction of the magnetic field at the common centre of the two coils, if they carry currents equal to I and \[\sqrt{3}\] I respectively.

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Solution

Magnetic field at the centre of the coils due to coil P, having current I is 

\[B_P = \frac{\mu_0 I}{2R}\]
And magnetic field due to coil Q having current 
\[\sqrt{3}I\] is \[B_Q = \frac{\mu_0 \sqrt{3}I}{2R}\] 
Since both coils are inclined to each other at an angle of 90°, the magnitude of their resultant magnetic field at the common centre will be
\[B = \sqrt{{B_P}^2 + {B_Q}^2} = \frac{\mu_0 I}{2R}\sqrt{1 + 3} = \frac{\mu_0 I}{R}\]
The directions of BP and BQ are as indicated in the figure. The direction of the resultant field is at an angle θ given by
\[\theta = \tan^{- 1} \left( \frac{B_P}{B_Q} \right) = \tan^{- 1} \left( \frac{1}{\sqrt{3}} \right) = 30°\]
Hence, the direction of the magnetic field will be at an angle 30° to the plane of loop P.
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2015-2016 (March) Foreign Set 2

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