Question

A dip circle shows an apparent dip of ${60}^{\circ }$ at a place where the true dip is ${45}^{\circ }$. If the dip circle is rotated by ${90}^{\circ }$, what will be the apparent dip?
A) ${\cos }^{-1}\sqrt{{\displaystyle \frac{2}{3}}}$
B) ${\tan }^{-1}\sqrt{{\displaystyle \frac{2}{3}}}$
C) ${\sin }^{-1}\sqrt{{\displaystyle \frac{2}{3}}}$
D) ${\cot }^{-1}\sqrt{{\displaystyle \frac{2}{3}}}$

Answer 1

Hint: Dip circles are used to determine the angle between the sky and the magnetic field of the Earth. Dip circle, also called Dip Needle is an instrument for measuring the inclination or dip of the Earth's magnetic field. When the instrument is positioned with the circle plane in the Earth's magnetic meridian, the needle points in the direction of the Earth's magnetic field.

Complete step by step answer:
The dip angle is the angle of the overall field vector with respect to the horizontal plane and is positive for vectors below the plane. It is a complement to the normal polar angle of the spherical coordinates.

Apparent dip is the name of any dip measured in a vertical plane not perpendicular to the strike line. True dip can be measured from the observable dip using trigonometry if the strike is known. Geological cross-sections use an obvious dip when drawn at a certain angle that is not perpendicular to the effect.

The dip at the position is determined by the dip circle. It consists of a magnetised needle capable of spinning around a horizontal axis in a vertical plane. The angle of the horizontal needle is called the Apparent Dip.
If ${\theta }_1$ and ${\theta }_2$ are the angles of sip in two vertical planes at right angle to each other and $\theta$ is the true dip then:
${\cot }^2\theta ={\cot }^2{\theta }_1+{\cot }^2{\theta }_2$
Therefore,
$$\begin{gathered} {\cot }^2{\theta }_2={\cot }^2\theta -{\cot }^2{\theta }_1 \\ ={\cot }^2{45}^{\circ }-{\cot }^2{60}^{\circ } \\ =1-{\displaystyle \frac{1}{3}} \\ ={\displaystyle \frac{2}{3}} \end{gathered}$$
Hence,
${\theta }_2={\cot }^{-1}\sqrt{{\displaystyle \frac{2}{3}}}$

Thus, option (D) is correct.

Note: Here we have to observe which angle is ${\theta }_1$ and which angle is ${\theta }_2$. Also, we may be confused as to which angle is given to find, so we have to be careful.