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

Answer: (B)

$\sqrt{5}$ B

Answer 2

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, $B=\sqrt{B_{P}^{2}+B_{Q}^{2}}$ $=\sqrt{{{(2B)}^{2}}+{{B}^{2}}}$ $=\sqrt{5{{B}^{2}}}$ $=\sqrt{5B}$