Two short bar magnets P and Q are arranged such that their centres are on the X-axis and are separated by a large distance. The magnetic axes of P and Q are along X and Y axes respectively. At a point R, midway between their centres, if B is the magnitude of induction due to Q, the magnitude of total induction at R due to the both magnets is:

Question

Two short bar magnets P and Q are arranged such that their centres are on the X-axis and are separated by a large distance. The magnetic axes of P and Q are along X and Y axes respectively. At a point R, midway between their centres, if B is the magnitude of induction due to Q, the magnitude of total induction at R due to the both magnets is: (A) 3B (B)  √5 B (C)  (√5/2) B (D) B

Tardigrade Admin · Accepted Answer

The point R will be in end-on position with respect to one dipole and in broad-side on position with respect to the other.
So,

B_{P} = \frac{μ _{0}}{4 π} \frac{2 M _{1}}{r _{1}^{3}}

and

B_{Q} = \frac{μ _{0}}{4 π} \frac{M _{2}}{r _{2}^{3}}

Since,

M_{1} = M_{2} = M, r_{1} = r_{2} = r

and

B_{Q} = B

∴

B_{P} = 2 B

However, as

B_{P}

and

B_{Q}

are perpendicular to each other,

B = B_{P}^{2} + B_{Q}^{2}

= (2 B)^{2} + B^{2}

= 5 B^{2}

= 5 B

3B

$5$ B

$\frac{5}{2}$ B

B

3B

5​ B

25​​ B

B

$5$ B

$\frac{5}{2}$ B