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Question: The real angle of dip, if a magnet is suspended at an angle \({30^0}\) to the magnetic meridian and ...

The real angle of dip, if a magnet is suspended at an angle 300{30^0} to the magnetic meridian and the dip needle makes an angle of 450{45^0} with horizontal is:
(A) tan⁡−1(32){\tan ^{ - 1}}\left( {\dfrac{{\sqrt 3 }}{2}} \right)
(B) tan⁡−1(3){\tan ^{ - 1}}\left( {\sqrt 3 } \right)
(C) tan⁡−1(12){\tan ^{ - 1}}\left( {\dfrac{1}{2}} \right)
(D) tan⁡−1(23){\tan ^{ - 1}}\left( {\dfrac{2}{{\sqrt 3 }}} \right)

Explanation

Solution

Remember the different value of tan⁡θ\tan \theta and cos⁡θ\cos \theta for solving trigonometry. The angle of dip is the total field vector mix with respect to the horizontal plane. It is positive for the vector below the plane. It is the complement of the usual polar angle of spherical coordinates. Angle of dip is also known as the magnetic dip and is defined as the angle that is made by the earth’s magnetic fields like with the horizontal. The angle of dip varies from point to point by providing the information related to the movement of the earth’s magnetic field.

Complete step by step answer:
The angle of dip is said to be positive when the magnetic field points downwards. When the magnetic field points upwards, the angle of dip is said to be negative. The angle of dip is 0∘0^\circ when the dip needle rests horizontally while the angle dip is 90∘90^\circ when the dip needle rests vertically.
Angle of dip δ=45∘\delta = {45^\circ }
∴tan⁡δ1=tan⁡δcos⁡θ\therefore \tan {\delta ^1} = \dfrac{{\tan \delta }}{{\cos \theta }}
=tan⁡45∘cos⁡45∘= \dfrac{{\tan 45^\circ }}{{\cos 45^\circ }}
=132= \dfrac{1}{{\dfrac{{\sqrt 3 }}{2}}}
δ=tan⁡−123\delta = {\tan ^{ - 1}}\dfrac{2}{{\sqrt 3 }}
Hence the answer is tan⁡−1(23){\tan ^{ - 1}}\left( {\dfrac{2}{{\sqrt 3 }}} \right)

So, the correct answer is “Option D”.

Note:
The angle between the magnetic meridian and geographic meridian at a place is called magnetic declination. Magnetic inclination or angle of dip is the angle between the direction of the total magnetic field of work and the horizontal line in the magnetic meridian. Angle of dip is used for measuring and correcting the magnetic compass error. It also has its uses in the geological field and in mapping. The angle of dip is 0∘0^\circ when the dip needle rests horizontally. While the angle of dip is 90∘90^\circ the dip needle rests vertically. When horizontal components and vertical components of earth’s magnetic field are equal the angle of dip is equal to 45∘45^\circ angle of dip at equator is equal to0∘0^\circ because the magnetic line of forces are perfectly horizontal such that even the magnetic needle is horizontal there. The calculus is based on trigonometry and algebra. The fundamental trigonometric functions like sine and cosine are used to describe the sound and light waves. Trigonometry is used in oceanography to calculate heights of waves and tides in oceans.