A compass needle oscillates 20 times per minute at a place where the dip is 30° and 30 times per minute where the dip is 60°. The ratio of total magnetic field due to the earth at two places respectively is 4/√x. The value of x is

Solution

Physics sir · 4 steps

A compass needle swings horizontally, so only the horizontal field B cos δ sets its frequency.

1. Frequency and field. For a magnet oscillating in a field, T = 2π√(I/mB), so f² ∝ B. A compass needle moves in the horizontal plane, so the field that matters is B_H = B cos δ, where δ is the dip.

   f₁²/f₂² = (B₁cos δ₁)/(B₂cos δ₂)

2. Put in numbers. The left side is (20/30)² = 4/9. On the right, (cos 30°)/(cos 60°) = (√3/2)/(1/2) = √3, so 4/9 = √3 B₁/B₂.
3. Solve the ratio. B₁/B₂ = 4/9√3 = 4/√(81 × 3) = 4/√243.
4. So. Matching with 4/√x gives x = 243.

The common slip: 27. Using B sin δ, the vertical component, instead of B cos δ gives B₁/B₂ = 4√3/9 = 4/√27, so 27, not 243.

The idea to keep. Frequency squared tracks the field component the needle actually moves in. For a horizontal compass that is B cos δ, never B itself or B sin δ.

Answer from the NTA final answer key. Solution by PrepWiser.