Question

Two magnets $A$ and $B$ are identical and these are arranged as shown in the figure. Their length is negligible in comparison with the separation between them. A magnetic needle is placed between the magnets at point $P$ which gets deflected through an angle $\theta$ under the influence of magnets. The ratio of distances $d_1$ and $d_2$ will be :

• A. $(2\tan \theta {)}^{\frac{1}{3}}$
• B. $(2\tan \theta {)}^{-\frac{1}{3}}$
• C. $(2\cot \theta {)}^{\frac{1}{3}}$
• D. $(2\cot \theta {)}^{-\frac{1}{3}}$

Answer 1

Answer: (C)

$(2\cot \theta {)}^{\frac{1}{3}}$

The magnetic needle will align along the total magnetic field.
$\tan \theta =\frac{B_A}{B_B}$
$B_A=\frac{2{\mu }_{0}m}{4\pi d{{}_1}^2},B_B=\frac{{\mu }_{0}m}{4\pi d{{}_2}^2},$
$\tan \theta =\frac{(\frac{2}{d{{}_1}^3})}{d{{}_2}^3}$
${\left(\frac{d_1}{d_2}\right)}^3=\frac{2}{\tan \theta }$
$\frac{d_1}{d_2}={\left(2\cot \theta \right)}^{1/3}$