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. $ \sqrt{5} $ B
• C. $ \frac{\sqrt{5}}{2} $ B
• D. B

Answer 1

The Correct Option is B

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} $