What is the formula for the magnetic field induction at any general point ‘p’

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$B = \frac{μ_{0}}{4 π} \frac{2 M}{d^{3}} (\sqrt{1 + 3 \cos^{2} θ})$

$B = \frac{μ_{0}}{4 π} \frac{2 M}{d^{3}} (\sqrt{1 + 3 \sin^{2} θ})$

$B = \frac{μ_{0}}{4 π} \frac{M}{d^{3}} (\sqrt{1 + 3 \sin^{2} θ})$

$B = \frac{μ_{0}}{4 π} \frac{M}{d^{3}} (\sqrt{1 + 3 \cos^{2} θ})$

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Detailed Solution

Magnetic induction at point ‘p’ due to magnet $N_{1} S_{1}$ of moment
$M \cos θ$ is $B = \frac{μ_{0}}{4 π} \frac{M \cos θ}{d^{3}}$ ---(1) (Point lies on axial line)
Magnetic induction at point ‘p’ due to magnet $N_{1} S_{1}$ of moment
$M \cos θ$ is $B = \frac{μ_{0}}{4 π} \frac{M \cos θ}{d^{3}}$ ------( 2 ) (Point lies on equatorial line)
These two Magnetic inductions are perpendicular to each other. The resultant Magnetic induction at p is given by
$B = \sqrt{B_{1}^{2} + {B_{2}}^{2}} = \frac{μ_{0}}{4 π} \frac{M}{d^{3}} (\sqrt{1 + 3 \cos^{2} θ})$