question_answer

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)  \[\sqrt{5}\]B

C)   \[\frac{\sqrt{5}}{2}\]B                                 

D)  B

Correct Answer: B

Solution :

                 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,                          \[{{B}_{P}}=\frac{{{\mu }_{0}}}{4\pi }\frac{2{{M}_{1}}}{r_{1}^{3}}\] and                        \[{{B}_{Q}}=\frac{{{\mu }_{0}}}{4\pi }\frac{{{M}_{2}}}{r_{2}^{3}}\] Since,    \[{{M}_{1}}={{M}_{2}}=M,\,\,{{r}_{1}}={{r}_{2}}=r\]and \[{{B}_{Q}}=B\] \[\therefore \]  \[{{B}_{P}}=2B\] However, as \[{{B}_{P}}\]and \[{{B}_{Q}}\]are perpendicular to each other, \[B=\sqrt{B_{P}^{2}+B_{Q}^{2}}\] \[=\sqrt{{{(2B)}^{2}}+{{B}^{2}}}\] \[=\sqrt{5{{B}^{2}}}\] \[=\sqrt{5B}\]