Question

A dip needle lies initially in the magnetic meridian when it shows an angle of dip at a place. The dip circle is rotted through an angle \(\mathrm{x}\) in the horizontal plane and then it shows an angle of dip \(\theta^{\prime}\). Then \(\left[\left(\tan \theta^{\prime}\right) /(\tan \theta)\right]\) is (a) \([1 /(\cos x)]\) (b) \([1 /(\sin x)]\) (c) \([1 /(\tan \mathrm{x})]\) (d) \(\cos \mathrm{x}\)

Short answer

The short answer is \(\left[\frac{\tan \theta^{\prime}}{(\tan \theta)}\right] = \left[\frac{1}{\cos x}\right]\), which corresponds to option (a).

Step-by-step solution

Understanding the problem

Before we can begin to solve this problem, let's make sure we understand each term. The angle of dip (θ) measures the angle of Earth's magnetic field lines downward from the horizontal, at a particular location. In this exercise, the dip needle lies initially in the magnetic meridian, meaning it's pointing in the direction of Earth's magnetic field (north-south direction). Its angle of dip is given as θ. After rotating the needle by an angle x in the horizontal plane, it points to a new direction, and it shows a final angle of dip, θ'. We are to find the relationship between the final angle of dip and the initial angle of dip.

Express θ' in terms of θ and x

To do this, we'll use some trigonometry. We'll start by writing the following expression: \( \tan \theta^{\prime} = \frac{\tan \theta}{f(x)} \) This equation expresses the relationship between the final angle of dip (θ') and the initial angle of dip (θ) in terms of some unknown function of x, f(x).

Examine given options for values of f(x)

Now, we will examine each of the four given choices in order to determine which of them is the correct function f(x): (a) \( f(x) = \cos x \) (b) \( f(x) = \sin x \) (c) \( f(x) = \tan x \) (d) \( f(x) = \frac{1}{\cos x} \)

Use the definition of dip angle to analyze the effect of rotation angle x

From the definition of the dip angle, we know that the horizontal component of the magnetic field remains unchanged. As the dip needle rotates, it measures the angle of dip relative to the new direction. When the dip circle rotates through an angle x, the horizontal component of the magnetic field in the new direction is the ratio of horizontal component relative to the rotated x: \( H' = H\cos x \) Where H is the horizontal component of the magnetic field initially and H' is the horizontal component of the magnetic field after rotating. Now, we can define the tangent of the angle θ' as follows: \( \tan \theta^{\prime} = \frac{V'}{H'} \) where V' is the vertical component of the magnetic field. In the initial direction, we can define the tangent of the angle θ as follows: \( \tan{\theta} = \frac{V}{H} \) Where V is the vertical component of the magnetic field. As the dip needle rotates, the vertical component V remains unchanged, hence V'=V. Therefore, we can write the following equation: \( \frac{\tan \theta^{\prime}}{\tan \theta} = \frac{H}{H'} = \frac{1}{\cos x} \)

Conclusion

From the above analysis, we can conclude that the correct answer is: \( \left[\frac{\tan \theta^{\prime}}{(\tan \theta)}\right] = \left[\frac{1}{\cos x}\right] \) Hence, the correct answer is (a).