# A dip circle is so set that its needle moves freely in the magnetic meridian. In this position, the angle of dip is 40°. Now the dip circle is rotated so that the plane in which the needle moves makes an angle of 30° with the magnetic meridian. In this position, the needle will dip by an angle

A. 40°

B. 30°

C. More than 40°

D. Less than 40°

Answer

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**Hint:**Use the relation between apparent angle, true angle and angle of dip with meridian. Substitute the given values directly into the formula and find the true angle of dip.

**Formula used:**

$\cot ^{ 2 }{ \theta \quad =\quad \cot ^{ 2 }{ { \theta }_{ 1 }+\quad \cot ^{ 2 }{ { \theta }_{ 2 } } } }$

**Complete answer:**

Let $ { \theta }_{ 1 }$ be the apparent angle of dip.

$ { \theta }_{ 2 }$ be the angle of dip with magnetic meridian.

$ \theta$ be the true angle of dip.

Given:

^{ }${ \theta }_{ 2 }$= 30° and

${ \theta }_{ 1 }$= 40°

Now, we have the relation between apparent angle of dip and true angle of dip,

$\cot ^{ 2 }{ \theta \quad =\quad \cot ^{ 2 }{ { \theta }_{ 1 }+\quad \cot ^{ 2 }{ { \theta }_{ 2 } } } }$

$\Rightarrow \cot { \theta \quad =\sqrt { \cot ^{ 2 }{ { \theta }_{ 1 }+\cot ^{ 2 }{ { \theta }_{ 2 } } } } }$

By substituting the values in above equation we get,

$ \cot { \theta \quad =\sqrt { \cot ^{ 2 }{ 40°+ } \cot ^{ 2 }{ 30° } } }$

$\Rightarrow \cot { \theta \quad =\sqrt { { 1.19 }^{ 2 }+{ \sqrt { 3 } }^{ 2 } } }$

$\Rightarrow \cot { \theta \quad =\sqrt { 1.42+3 } }$

$\Rightarrow \cot { \theta \quad =\quad 4.42 }$

$\Rightarrow \quad \theta =\quad 25°$

Therefore, the true angle of dip is less than 40°.

**So, the correct answer is “Option D”.**

**Note:**

There’s an alternate method to solve this problem. The alternate method is shown below.

Formula used to find the true dip is given by,

$\tan { { \theta }_{ 1 }=\dfrac { \tan { \theta } }{ cos{ \theta }_{ 2 } } }$

By substituting the values in above equation we get,

$\tan { 40°=\dfrac { \tan { \theta } }{ cos30° } }$

$\Rightarrow \tan { \theta =\quad \tan { 40° } \times \cos { 30° } }$

$\Rightarrow \tan { \theta =\quad 0.84\quad \times \quad 0.87 }$

$\Rightarrow \quad \theta =\tan ^{ -1 }{ 0.73 }$

$\Rightarrow \quad \theta =\quad 36.13°$

Therefore, the true dip is less than 40°.

Hence, the correct answer is option D i.e. less than 40°.

Last updated date: 03rd Jun 2023

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