Question

If the earth's magnetic field at a certain place has a horizontal component of $ 5 $ gauss and a total field of $ 13 $ gauss, then the angle of dip at that place is:

• A. $ \tan^{-1}\left(\frac{5}{12}\right) $
• B. $ \tan^{-1}\left(\frac{5}{13}\right) $
• C. $ \tan^{-1}\left(\frac{12}{5}\right) $
• D. $ \tan^{-1}\left(\frac{13}{5}\right) $

Hint:

The angle of dip is determined using the ratio of the vertical component to the horizontal component of the earth's magnetic field: $ \tan \delta = \frac{B_V}{B_H} $.

Answer 1

The Correct Option is C

Step 1: Understanding Earth's Magnetic Field Components
The earth's magnetic field can be resolved into two components:
1. Horizontal Component ($ B_H $): The component parallel to the surface of the earth.
2. Vertical Component ($ B_V $): The component perpendicular to the surface of the earth. The total magnetic field ($ B_T $) is given by: $$ B_T = \sqrt{B_H^2 + B_V^2} $$ The angle of dip ($ \delta $) is defined as the angle between the total magnetic field vector and the horizontal plane. It is related to the components by: $$ \tan \delta = \frac{B_V}{B_H} $$ Step 2: Given Values
Horizontal component: $ B_H = 5 $ gauss
Total field: $ B_T = 13 $ gauss
Using the Pythagorean theorem: $$ B_T = \sqrt{B_H^2 + B_V^2} $$ Substitute the given values: $$ 13 = \sqrt{5^2 + B_V^2} $$ $$ 13 = \sqrt{25 + B_V^2} $$ Square both sides: $$ 169 = 25 + B_V^2 $$ $$ B_V^2 = 144 \quad \Rightarrow \quad B_V = \sqrt{144} = 12 \text{ gauss} $$ Step 3: Calculate the Angle of Dip
The angle of dip is given by:
$$ \tan \delta = \frac{B_V}{B_H} = \frac{12}{5} $$ Thus: $$ \delta = \tan^{-1}\left(\frac{12}{5}\right) $$ Step 4: Analyzing Options Option (1): $ \tan^{-1}\left(\frac{5}{12}\right) $
Incorrect, as this corresponds to the ratio of the horizontal component to the vertical component. Option (2): $ \tan^{-1}\left(\frac{5}{13}\right) $
Incorrect, as this does not match the correct ratio. Option (3): $ \tan^{-1}\left(\frac{12}{5}\right) $
Correct, as it matches the calculated ratio $ \frac{B_V}{B_H} = \frac{12}{5} $. Option (4): $ \tan^{-1}\left(\frac{13}{5}\right) $
Incorrect, as this does not correspond to the correct ratio.