A short magnetic dipole of moment M is at the origin with its magnetic moment along +x. At a distant point, the position vector makes an angle θ with +x. If the magnetic field at that point is perpendicular to the magnetic moment, θ is:
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Question

A short magnetic dipole of moment M is at the origin with its magnetic moment along +x. At a distant point, the position vector makes an angle θ with +x. If the magnetic field at that point is perpendicular to the magnetic moment, θ is:

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G1 Physics · Accepted Answer

The magnetic field of a dipole is B⃗ = (μ₀/4πr³)[3(M⃗·r̂)r̂ - M⃗]. For B⃗ to be perpendicular to M⃗, B⃗·M⃗ = 0. Thus, 3(M⃗·r̂)² - M² = 0. Since M⃗·r̂ = M cosθ, 3M²cos²θ - M² = 0. Therefore, 3cos²θ = 1 ⇒ cosθ = 1/√3 (for 0°≤θ≤90°). Hence, θ = cos⁻¹(1/√3).