Question

At a place dip angle is known to be \[45{}^\circ \]. The apparent dip when dip circle is placed at an angle of \[60{}^\circ \]with the magnetic meridian will be
\[A.\,{{\tan }^{-1}}\left( 2 \right)\]
\[B.\,{{\tan }^{-1}}\left( \dfrac{1}{2} \right)\]
\[C.\,{{\tan }^{-1}}\left( \sqrt{2} \right)\]
\[D.\,{{\tan }^{-1}}\left( \dfrac{1}{\sqrt{2}} \right)\]

Answer 1

Hint: The problem is based on using the formulae that relate the magnetic dip and the magnetic declination. The actual and the apparent dip values are calculated and then compared to obtain the expression in terms of the apparent dip. Finally, the given values are substituted in the obtained expression to find the required value.

Formula used:
\[\tan \rho =\dfrac{V}{H}\]

Complete step-by-step answer:
From the data, we have the data as follows.
The true value of the dip is,
\[\rho =45{}^\circ \]
The magnetic dip is the result of the tendency of a magnet to align itself with the lines of the magnetic field. A dip needle measures the inclination or the dip of the Earth’s magnetic field.

The magnetic declination is,
\[\theta =60{}^\circ \]
The magnetic declination is the angle on the horizontal plane between the magnetic north (the direction of the earth’s magnetic field lines) and the true north, that is, the geographical north pole.

First, let us obtain the expression for the true dip.
So, we have,
\[\tan \rho =\dfrac{V}{H}\] …… (1)
Where V is the vertical component and H is the horizontal component.

Now let us obtain the expression for the apparent dip.
So, we have,
\[\tan \rho '=\dfrac{V}{H\cos \theta }\] …… (2)
Where V is the vertical component and \[H\cos \theta \]is the horizontal component, because of the inclination.

Now compare the equations (1) and (2) and obtain the expression in terms of the apparent dip.
So, we get,
\[\tan \rho '=\dfrac{\tan \rho }{\cos \theta }\]

Substitute the given values in the above equation, so, we get,
\[\tan \rho '=\dfrac{\tan 45{}^\circ }{\cos 60{}^\circ }\]
Substitute the values of the angles.
\[\begin{align}
  & \tan \rho '=\dfrac{1}{{}^{1}/{}_{2}} \\
 & \Rightarrow \tan \rho '=2 \\
\end{align}\]

Therefore, the value of the apparent dip is given as, \[\rho '={{\tan }^{-1}}(2)\]

As the value of the apparent dip is\[{{\tan }^{-1}}(2)\],

So, the correct answer is “Option A”.

Additional Information: The dip of the magnetic needle is called the magnetic inclination. This dip is the angle made by the magnetic needle of the magnetic meridian with the horizontal plane at the specific location.
At the magnetic equator, the magnetic dip measures \[0{}^\circ \]and at each of the magnetic poles, the magnetic dip measures \[90{}^\circ \].

Note: The things to be on your finger-tips for further information on solving these types of problems are: Instead of the dip angle, the question may be asked for magnetic inclination, so both these are meant to be the same.