Question

Two short magnets with equal dipole moments $ M $ are placed perpendicularly at their centers. The magnitude of the magnetic field at a distance $ d $ from the center on the bisector of the right angle is:

• A. \( \frac{\mu_0}{4\pi} \cdot \frac{2\sqrt{2}M}{d^3} \)
• B. \( \frac{\mu_0}{4\pi} \cdot \frac{5M}{d^3} \)
• C. \( \frac{\mu_0}{4\pi} \cdot \frac{2M}{d^3} \)
• D. \( \frac{\mu_0}{4\pi} \cdot \frac{10M}{d^3} \)

Hint:

Use vector addition of magnetic dipole fields when dipoles are perpendicular and field is along bisector.

Answer 1

The Correct Option is A

Magnetic field on axial line of a dipole: \[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{r^3} \] If two identical dipoles are placed perpendicularly at center and we are at angle bisector (i.e., 45° from each), the vector sum of both dipole fields: \[ B_{net} = \sqrt{B^2 + B^2} = B\sqrt{2} \] Each contributes \( \frac{\mu_0}{4\pi} \cdot \frac{2M}{d^3} \) \[ B_{net} = \sqrt{2} \cdot \frac{\mu_0}{4\pi} \cdot \frac{2M}{d^3} = \frac{\mu_0}{4\pi} \cdot \frac{2\sqrt{2}M}{d^3} \]