# A magnetic compass needle oscillates 30 times per minute at a place where the dip is $45^{\circ}$, and $40$ times per minute where the dip is $30^{\circ}$. If $B_{1}$ and $B_{2}$ are respectively the total magnetic field due to the earth at the two places, then the ratio $B_{1}/B_{2}$ is best given by : Option 1) $2.2$ Option 2) $0.7$ Option 3) $1.8$ Option 4) $3.6$

Compass needle 1 $\rightarrow$

Compass needle 2$\rightarrow$

$I_{1}=\frac{1}{2\pi}\sqrt{\frac{rB_{1}\cos45^{\circ}}{I}}$

$I_{2}=\frac{1}{2\pi}\sqrt{\frac{rB_{2}\cos30^{\circ}}{I}}$

$\Rightarrow \frac{I_{1}}{I_{2}}=\sqrt{\frac{B_{1}\cos45^{\circ}}{B_{2}\cos30^{\circ}}}$

$\Rightarrow \frac{B_{1}}{B_{2}}=0.7$

Option 1)

$2.2$

Option 2)

$0.7$

Option 3)

$1.8$

Option 4)

$3.6$

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