First slide

Magnetic dipole in a uniform magnetic field

Question

Two identical magnetic dipoles of magnetic moments $1.0 A - m^{2}$ each, placed at a separation of 2m with their axes perpendicular to each other, The resultant magnetic field at a point mid-way between the dipoles is

Difficult

A

$5 \times 10^{- 7} T$
B

$\sqrt{5} \times 10^{- 7} T$
C

$10^{- 7} T$
D

$2 \times 10^{- 7} T$

Solution

$\begin{array}{l} |\vec{P_{1}}| = |\vec{P_{2}}| = P \\ B_{1} = \frac{μ_{0}}{4 π} . \frac{P_{1}}{{(r / 2)}^{3}} = \frac{μ_{0}}{4 π} \frac{8 P_{1}}{r^{3}} = 10^{- 7} T \\ B_{2} = \frac{μ_{0}}{4 π} \times \frac{2 P_{1}}{{(r / 2)}^{3}} = \frac{μ_{0}}{4 π} \frac{16 P_{2}}{r^{3}} = 2 \times 10^{- 7} T \\ ∴ B = \sqrt{{B_{1}}^{2} + {B_{2}}^{2}} = \sqrt{5} \times 10^{- 7} T \end{array}$

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