Q. Two identical bar Magnets each having Magnetic Moment of ‘M’ are kept at a distance of 2d with their axes perpendicular to each other in a horizontal plane. The Magnetic induction at midway between them is

(a) $ \displaystyle \frac{\mu_0}{4\pi}\sqrt2 \frac{M} { d^3 }$

(b) $ \displaystyle \frac{\mu_0}{4\pi}\sqrt3 \frac{M} { d^3 }$

(c) $ \displaystyle \frac{\mu_0}{4\pi}\frac{M} { d^3 }$

(d) $ \displaystyle \frac{\mu_0}{4\pi}\sqrt5 \frac{M} { d^3 }$

Ans: (d)

Sol: For one magnet Point lies on axial line and for other point lies on equitorial line .

Magnetic induction at a point on axial line due to a bar magnet

$ \displaystyle B_1 = \frac{\mu_0}{4\pi} \frac{2 M}{d^3}$ …(i)

Magnetic induction at a point on equitorial line due to a bar magnet

$ \displaystyle B_2 = \frac{\mu_0}{4\pi} \frac{ M}{d^3}$ …(ii)

From (i) & (ii)

$ \displaystyle B_2 = \frac{B_1}{2} $

Net magnetic induction is

$\displaystyle B = \sqrt{B_1^2 + B_2^2 }$

$\displaystyle B = \sqrt{(2B_2)^2 + B_2^2 }$

$ \displaystyle B = \sqrt{5 B_2^2 }$

$latex \displaystyle B = \sqrt{5} B_2 $

$ \displaystyle B = \sqrt{5} \frac{\mu_0}{4\pi} \frac{ M}{d^3} $