Two identical short bar magnets, each of magnetic moment M, have the same centre and mutually perpendicular magnetic axes. At a point P at distance r on the axial line of the first magnet, the magnitude of the resultant magnetic field is:
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Question

Two identical short bar magnets, each of magnetic moment M, have the same centre and mutually perpendicular magnetic axes. At a point P at distance r on the axial line of the first magnet, the magnitude of the resultant magnetic field is:

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G1 Physics · Accepted Answer

Let C = μ₀M/(4πr³). For the first magnet, P is on its axial line: B₁ = 2C. For the second magnet, P lies on its equatorial line: B₂ = C. The axial field of the first magnet is along its magnetic moment, while the equatorial field of the second magnet is opposite to its magnetic moment. Since the two magnetic moments are perpendicular, B₁ and B₂ are also perpendicular. Therefore, B = √(B₁²+B₂²) = √[(2C)²+C²] = √5 C. Hence, B = (μ₀M/4πr³)√5.