Three identical thin bar magnets each of moment M are placed along three adjacent sides of a regular hexagon …..

Q. Three identical thin bar magnets each of moment M are placed along three adjacent sides of a regular hexagon as shown in figure. The resultant magnetic moment of the system is

(a) M

(b) M√3

(c) M√2

(d) 2M

Ans: (d)

Sol: $\displaystyle \vec{M_R} = M \hat{i} + (M cos 60 \hat{i} + M sin 60 \hat{j}) + (M cos 60 (-\hat{i}) + M sin60 \hat{j})$

$\displaystyle \vec{M_R} = M \hat{i} + 2 M sin 60 \hat{j}$

$\displaystyle \vec{M_R} = M \hat{i} + 2 M \times \frac{\sqrt{3}}{2} \hat{j}$

$\displaystyle \vec{M_R} = M \hat{i} + \sqrt{3} M \hat{j}$

$\displaystyle M_R = \sqrt{M^2 + (\sqrt{3}M)^2}$

= 2 M