Question

If a magnet is suspended at angle $30°$ to the magnetic meridian, the dip needle makes an angle of $45°$ with the horizontal. The real dip is :
A.${tan}^{-1}(\sqrt{{3}/{2}})$
B.${tan}^{-1}(\sqrt 3)$
C.${tan}^{-1}(\sqrt{3}/{2})$
D.${tan}^{-1}(2\sqrt{3})$

Answer 1

Hint: Earth’s magnetic field has two components i.e. vertical and horizontal components. The angle made between the horizontal and vertical component is called angle of dip. As an angle is formed due to the suspension of the magnet with the magnetic direction, the true dip changes and hence we get the apparent dip. So, this problem can be solved using the relation between apparent dip and real dip. Substitute the values in that equation and find the value of real dip.

Complete step-by-step solution:
Given: Apparent dip $\theta$ = $45°$
           Angle made by dip circle with the meridian $\alpha$ = $30°$
Let V and H be the vertical and horizontal components of Earth’s magnetic field.
      $\delta$ be the real dip
Thus, angle of dip is given by,
 $\tan { \delta =\dfrac { V }{ H } }$ …(1)
As the angle is made by a dip circle with the meridian vertical component remains the same while the horizontal component changes.
Thus, $H=H\cos { \alpha } $
Now, angle of dip becomes,
$\tan {45°} =\dfrac { V }{ H cos{\theta} }$
$\Rightarrow \dfrac { V }{ H } =\tan { 45° } \times \cos { 30° }$
$\Rightarrow \dfrac { V }{ H } =1\times \dfrac { \sqrt { 3 } }{ 2 }$
$\Rightarrow \dfrac { V }{ H } =\dfrac { \sqrt { 3 } }{ 2 }$ …(2)
Substituting equation. (2) in equation. (1) we get,
$\tan { \delta = } \dfrac { \sqrt { 3 } }{ 2 }$
$\Rightarrow \delta =\tan ^{ -1 }{ \dfrac { \sqrt { 3 } }{ 2 } }$
Thus, the real dip is $\tan ^{ -1 }({ \dfrac { \sqrt { 3 } }{ 2 } })$

Hence, the correct answer is option C i.e. $\tan ^{ -1 }({ \dfrac { \sqrt { 3 } }{ 2 } })$.

Additional Information:
If there are two apparent dips at right angle to each other then the relation is given as,
${cot }^{2}\theta=\sqrt {{{cot}^{2}\theta _{1}}+{{cot}^{2}\theta_{2}}}$
Where, $\theta$ is the true angle
             ${\theta}_{1} and {\theta}_{2}$ are the apparent angles.
In that case, true dip is equal to the angle made by the dip circle with the meridian.

Note:
Dip is always between $0°$ and $90°$. Angle at the magnetic equator is $0°$ while that at the magnetic poles is $90°$.
Apparent dip can never be greater than real dip. Thus, every time you calculate real dip or apparent dip you can verify your answer by checking if the real dip is greater than apparent dip or not.
If after calculation you get apparent dip as negative, then take the absolute value of dip.