Question

If $ \theta_1 $ and $ \theta_2 $ be the apparent angles of dip observed in two vertical planes at right angles to each other, then the true angle of dip $ \theta $ is given by

• A. \( \cot^2 \theta = \cot^2 \theta_1 + \cot^2 \theta_2 \)
• B. \( \tan^2 \theta = \tan^2 \theta_1 + \tan^2 \theta_2 \)
• C. \( \cot^2 \theta = \cot^2 \theta_1 + \cot^2 \theta_2 \)
• D. \( \tan^2 \theta = \tan^2 \theta_1 - \tan^2 \theta_2 \)

Hint:

When the apparent angles of dip are given in two vertical planes at right angles to each other, use the formula \( \cot^2 \theta = \cot^2 \theta_1 + \cot^2 \theta_2 \) to find the true angle of dip.

Answer 1

The Correct Option is A

To analyze the relationship between magnetic field components in different planes, we examine two mutually perpendicular planes inclined to the magnetic meridian.

1. Problem Setup:
Consider two planes making angles θ and (90°-θ) with the magnetic meridian. The vertical component (B_V) remains constant, while the effective horizontal components differ:

• Plane 1: B₁ = B_Hcosθ
• Plane 2: B₂ = B_Hsinθ

 

2. Tangent Relationships:
For each plane, we express the angle of dip:

• Plane 1: cotθ₁ = (B_Hcosθ)/B_V
• Plane 2: cotθ₂ = (B_Hsinθ)/B_V

 

3. Deriving the Relationship:
Squaring and adding both expressions:

cot²θ₁ + cot²θ₂ = (B_H²cos²θ)/B_V² + (B_H²sin²θ)/B_V²
= B_H²(cos²θ + sin²θ)/B_V²
= B_H²/B_V²
= cot²θ

where θ is the angle of dip in the magnetic meridian plane.

 

Final Result:
The relationship between the angles is given by:
cot²θ₁ + cot²θ₂ = cot²θ

Answer 2

To solve for the true angle of dip \( \theta \), we must consider the geometry involved. The apparent angle of dip in a plane is the angle between the magnetic meridian and the horizontal plane in that particular vertical plane. Given that these planes are at right angles to each other, we can use this information to derive the relationship for the true angle of dip.

Let us analyze the relationship using the cotangent values of the angles:

From trigonometry, we know:

\(\cot \theta = \frac{1}{\tan \theta}\)

Using the relationships of the cotangent values for these angles, we can establish:

\(\cot^2 \theta = \cot^2 \theta_1 + \cot^2 \theta_2\)

This equation accounts for the perpendicular orientation of the planes and provides the correct relationship to calculate the true angle of dip \( \theta \) from the apparent angles \( \theta_1 \) and \( \theta_2 \).

Conclusion: The true angle of dip \( \theta \) is given by the equation \( \cot^2 \theta = \cot^2 \theta_1 + \cot^2 \theta_2 \).