A compass needle oscillates $20$ times per minute at a place where dip is $45^{0}$ and $30$ times per minute where the dip is $30^{0}$ . Ratio of the total magnetic field of earth at the two places respectively is

1.51

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1.83

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1.63

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1.23

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Solution

The correct option is B $1.83$
$\frac{B_{H_{2}}}{B_{H_{1}}} = \frac{T_{1}^{2}}{T_{2}^{2}} = \frac{B_{2} cos δ_{2}}{B_{1} cos δ_{1}} = \frac{T_{1}^{2}}{T_{2}^{2}}$
$\frac{B_{2}}{B_{1}} = \frac{T_{1}^{2} cos δ_{1}}{T_{2}^{2} cos δ_{2}}$
$= (\frac{3}{2})^{2} \frac{1 / \sqrt{2}}{\sqrt{3} / 2} = \frac{9}{4} \times \sqrt{\frac{2}{3}} = \frac{3 \sqrt{3}}{2 \sqrt{2}}$
$= 1.83$

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