Question

If the angle of dip at places A and B are 30° and 45° respectively, the ratio of horizontal component of earth's magnetic field at A to that at B will be

• A. \( \sqrt{2} : 1 \)
• B. \( 1 : \sqrt{2} \)
• C. \( \sqrt{2} : \sqrt{3} \)
• D. \( \sqrt{3} : \sqrt{2} \)

Hint:

The horizontal component of the magnetic field is proportional to the cosine of the angle of dip. Use trigonometric identities to solve such problems.

Answer 1

The Correct Option is D

Step 1: Horizontal component of magnetic field.
The horizontal component of the magnetic field \( H \) at a location is given by: \[ H = B \cos \delta \] where \( B \) is the magnetic field strength and \( \delta \) is the angle of dip.
Step 2: Apply to given angles.
At A, the angle of dip is 30°, and at B, it is 45°. The ratio of the horizontal components is: \[ \frac{H_A}{H_B} = \frac{B \cos 30^\circ}{B \cos 45^\circ} = \frac{\cos 30^\circ}{\cos 45^\circ} \]
Step 3: Simplify using known values.
Using \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) and \( \cos 45^\circ = \frac{1}{\sqrt{2}} \), we get: \[ \frac{H_A}{H_B} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{\sqrt{2}}} = \sqrt{3} : \sqrt{2} \]
Step 4: Conclusion.
The correct answer is (D).