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

Answer 1

The Correct Option is D

The problem asks us to find the expression for the true angle of dip when two apparent angles of dip, observed in two vertical planes at right angles to each other, are given. This is a standard problem from geomagnetism in Physics.

**Background Concept:**

The angle of dip (or magnetic inclination) is the angle made by the Earth's magnetic field with the horizontal plane. It indicates how steeply inclined the magnetic field lines are at any given location.

**Given:**

• $\theta_1$ and $\theta_2$ are the apparent angles of dip in two perpendicular vertical planes.

**To find:** The true angle of dip $\theta$.

**Theory and Derivation:**

1. Using trigonometric identities, the tangent of the true angle of dip is related to the apparent angles by this derived formula:
2. We know that the apparent angle of dip is represented in terms of tangent:
   $\tan \theta_1 = \frac{H}{V}$, $\tan \theta_2 = \frac{H}{V_2}$
3. Where H is the horizontal component, and V, V_2 are the vertical components in respective planes.
4. For the true dip angle $\theta$, we have:
   $\tan \theta = \frac{H}{V_{\text{true}}}$
5. The derived relation between the apparent angles $\theta_1, \theta_2$ and true angle of dip $\theta$ is:
   $\cot^2 \theta = \cot^2 \theta_1 + \cot^2 \theta_2$

**Conclusion:**

The correct expression for the true angle of dip is thus:

$\cot^2 \theta = \cot^2 \theta_1 + \cot^2 \theta_2$