Question

Two identical magnetic dipoles of magnetic moment $2A{m}^2$ are placed at a separation of $2m$ with their axes perpendicular to each other in air. The resultant magnetic field at a midpoint between the dipoles is

• A. $4\sqrt{5}\times 10^{-5}T$
• B. $2\sqrt{5}\times 10^{-5}T$
• C. $4\sqrt{5}\times 10^{-7}T$
• D. $2\sqrt{5}\times 10^{-7}T$
• E. $4\sqrt{2}\times {10}^{-7}T$

Answer 1

Answer: (D)

Let point $P$ be a midpoint between the dipoles. The point $P$ will be in end-on position with respect to one dipole and in broad-side on position with respect to the other.
$\therefore B_1=\frac{{\mu }_0}{4\pi }\frac{2m_1}{r_1^3}=\frac{10^{-7}\times 2\times 2}{{\left(1\right)}^3}=4\times 10^{-7}T$
and $B_2=\frac{{\mu }_0}{4\pi }\frac{m_2}{r_2^3}=\frac{10^{-7}\times 2}{{\left(1\right)}^3}=2\times 10^{-7}T$
As $B_1$ and $B_2$ are perpendicular to each other, therefore the resultant magnetic field at point $P$ is
$B=\sqrt{B_1^2+B_2^2}=\sqrt{{\left(4\times 10^{-7}\right)}^2+{\left(2\times 10^{-7}\right)}^2}$
$=10^{-7}\sqrt{16+4}=10^{-7}\sqrt{20}=2\sqrt{5}\times 10^{-7}T$