Answer

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**Hint:**The problem is grounded 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. Lastly, the given values are substituted in the acquired expression to find the required value.

Formula used:

$\Rightarrow \tan \rho = \dfrac{V}{H} $

**Complete step by step solution**

As per the given data, we have the data as follows.

The true value of the dip is,

$ \rho = {45^\circ } $

The magnetic dip is the consequence of the propensity of a magnet to align itself with the lines of the magnetic field. A dip needle deals with the inclination or the dip of the Earth’s magnetic field.

The magnetic declination is,

$ \theta = {45^\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 acquire the expression for the true dip.

So, we have,

$ \tan \rho = \dfrac{V}{H} $ ............ $ \left( 1 \right) $

where $ V $ is the vertical component and $ H $ is the horizontal component.

Now let us find the expression for the apparent dip.

So, we have,

$ \tan \rho ' = \dfrac{V}{{H\cos \theta }} $ ............ $ \left( 2 \right) $

where $ V $ is the vertical component and $ H\cos \theta $ is the horizontal component, because of the inclination.

Now compare the equations $ \left( 1 \right) $ and $ \left( 2 \right) $ to obtain the expression in terms of the apparent dip.

So, we get,

$ \tan \rho ' = \dfrac{{\tan \rho }}{{\cos \theta }} $

On Substituting the values in the above equation, so we get,

$ \tan \rho ' = \dfrac{{\tan {{45}^\circ }}}{{\cos {{45}^\circ }}} $

Substitute the values of the angles.

$ \tan \rho ' = \dfrac{1}{{\dfrac{1}{{\sqrt 2 }}}} $

$ \Rightarrow \tan \rho ' = \sqrt 2 $

Therefore, the tangent of the angle of apparent dip at the place is $ \sqrt 2 $ .

**So the correct option is (B) $ \sqrt 2 $ .**

**Additional information**

The dip of the magnetic needle is called the magnetic inclination. This dip is the angle made through 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 additional information on solving these types of problems are: in its place of the dip angle, the question may be requested for magnetic inclination, so both these are meant to be the same.

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