Following figures show the arrangement of bar magnets in different configurations. Each magnet has magnetic dipole moment . Which configuration has highest net magnetic dipole moment? [2014]

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Solution

Q.
Following figures show the arrangement of bar magnets in different configurations. Each magnet has magnetic dipole moment $\vec{m}$ . Which configuration has highest net magnetic dipole moment [2014]

1 (1)

2 (2)

3 (3)

4 (4)

Ans.
(3)

The direction of magnetic dipole moment is from south to north pole of the magnet.

In configuration (1),

$m_{net} = \sqrt{m^{2} + m^{2} + 2 m m \cos 90 °} = \sqrt{m^{2} + m^{2}} = m \sqrt{2}$

In configuration (2),

In configuration (3),

$m_{net} = \sqrt{m^{2} + m^{2} + 2 m m \cos 30 °} = \sqrt{2 m^{2} + 2 m^{2} (\frac{\sqrt{3}}{2})} = m \sqrt{2 + \sqrt{3}}$

In configuration (4),

$m_{net} = \sqrt{m^{2} + m^{2} + 2 m m \cos 60 °} = \sqrt{2 m^{2} + 2 m^{2} (\frac{1}{2})} = m \sqrt{3}$

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