Question

A short bar magnet has a length $2\tau$ and a magnetic moment $10A{m}^2$. Find the magnetic field at a point ${}^{\prime }{P}^{\prime }$ at a distance of $r=0.1m$ from the center of the magnet on the line. The line making an angle ${30}^0$ with the magnetic axis as shown in fig. Here$\tau$is negligible as compared to $r$.

• A. $\sqrt{\frac{7}{4}}\times {10}^{-3}T$
• B. $2\times {10}^{-3}T$
• C. $\sqrt{\frac{13}{4}}\times {10}^{-3}T$
• D. ${10}^{-3}T$

Answer 1

The correct option is C $\sqrt{\frac{13}{4}}\times {10}^{-3}T$

$\begin{array}{c}\text{Using formula}\to \\ B_1=\frac{\mu }{4\pi r^3}\left(2p\cos \theta \right)\\ B_2=\frac{\mu }{4\pi }\frac{\left(p\sin \theta \right)}{r^3}\end{array}$

$B=\sqrt{B_1^2+B_2^2}=\frac{\mu p}{4\pi r^3}\sqrt{4{\cos }^2\theta +{\sin }^2\theta }$

$\begin{array}{c}=\frac{{10}^{-7}\times 10}{{10}^{-3}}\sqrt{\frac{4\times 3}{4}+\frac{1}{4}}\\ =\frac{{10}^{-3}\sqrt{13}}{2}=\sqrt{\frac{13}{4}}\times {10}^{-3}\text{tesla}\end{array}$