Magnetic Properties of Matter

JEE Main level, single correct, medium

19 of 57

Question 19 of 57

A magnetic compass needle oscillates 3030 times per minute at a place where the dip is 45∘45^{\circ} and 4040 times per minute where the dip is 30∘30^{\circ}. If B1{B}_{1} and B2{B}_{2} are the net magnetic fields due to the earth at the two places respectively, then the ratio B1/B2{B}_{1}/{B}_{2} is approximately equal to

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Answer: (D) 0.70.7

Let f1{f}_{1} and f2{f}_{2} be the respective oscillation frequencies respectively for 45∘45^{\circ} and 30∘30^{\circ} dip. Then,
f1=12πMB1cos45∘I{f}_{1}=\dfrac{1}{2\pi }\sqrt{\dfrac{M{B}_{1}cos45^{\circ}}{I}}
f2=12πMB2cos⁡30∘I{f}_{2}=\dfrac{1}{2\pi }\sqrt{\dfrac{M{B}_{2}\cos 30^{\circ}}{I}}
f1f2=B1cos45∘B2cos30∘\dfrac{{f}_{1}}{{f}_{2}}=\sqrt{\dfrac{{B}_{1}cos45^{\circ}}{{B}_{2}cos30^{\circ}}}
⇒302402=B1cos45∘B2cos30∘\Rightarrow \dfrac{{30}^{2}}{{40}^{2}}=\dfrac{{B}_{1}cos45^{\circ}}{{B}_{2}cos30^{\circ}}
⇒B1B2=900×cos30∘1600×cos45∘\Rightarrow \dfrac{{B}_{1}}{{B}_{2}}=\dfrac{900\times cos30^{\circ}}{1600\times cos45^{\circ}}
⇒B1B2=0.688≈0.7\Rightarrow \dfrac{{B}_{1}}{{B}_{2}}=0.688\approx 0.7

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