Two short bar magnets of magnetic moments `m` each are arranged at the opposite corners of a square of side d such that their centres coincide with thecorners and their axes are parallel. If the like poles are in the same direction, the magnetic induction at any of the other corners of the square is

Options

A. `B = (mu_0)/(4pi) (Msqrt5)/(a^3)`
B. `B = (mu_0)/(4pi)(M sqrt2)/(a^3)`
C. `B = (mu_0)/(4pi) (M)/(a^3)`
D. `B = (mu_0)/(4pi) (3M)/(a^3)`

Answer

C

Explanation

The direction of magnetic moment of a magnet is from South Pole to North Pole. Let us find magnetic field intensity at point Q. Point Q lies on the axis of magnet placed at R. Direction of magnetic field intensity at a point on the axis of bar magnet is in the same direction as that of magnetic moment and magnitude is given by:
`B_1 = (mu_0)/(4pi) (2M)/(a^3)` , Direction of `B_1` is to along QR.
Point Q lies on the perpendicular bisector of bar magnet kept at corner P. Direction of magnetic field intensity due to a bar magnet at a point on its perpendicular bisector is opposite to its magnetic dipole moment and magnitude is given by
`B_2 = (mu_0)/(4pi ) (M)/(a^3) ` , Direction of `B_2` is along RQ.
Thus we can see in figure that net magnetic field is along QR and magnitude of net magnetic field can be written as follows:
`B = B_1 - B_2 = (mu_0)/(4pi) (2M)/(a^3) - (mu_0)/(4pi) (M)/(a^3)`
`B = (mu_0)/(4pi) (M)/(a^3)`