To understand the relationship between the angle of dip and the orientation of a dip needle, we need to delve into some fundamental concepts of magnetism and trigonometry. The angle of dip, also known as magnetic inclination, is the angle made by the Earth's magnetic field lines with the horizontal plane. When a dip needle is used, it aligns itself with the magnetic field, and the angle it makes with the horizontal can provide insights into the local magnetic environment.

Understanding the Dip Needle Setup

Imagine a dip needle, which is essentially a magnetized needle that can pivot freely. When it is placed in a magnetic field, it will align itself along the field lines. The angle it makes with the horizontal is crucial for determining the angle of dip at that location. If the dip needle is positioned in a vertical plane that is tilted relative to the magnetic meridian, the angle it makes with the horizontal will not be a straightforward representation of the angle of dip.

Analyzing the Geometry

Let’s break down the situation. The dip needle makes an angle of (90 - θ) with the horizontal, where θ is the angle of dip we want to find. The vertical plane in which the dip needle lies is at an angle φ from the magnetic meridian. This setup can be visualized as a right triangle where:

• The opposite side represents the vertical component of the magnetic field.
• The adjacent side represents the horizontal component of the magnetic field.

Using Trigonometric Relationships

To find the angle of dip θ, we can use the tangent function, which relates the opposite and adjacent sides of a right triangle. The tangent of the angle of dip can be expressed in terms of the angle φ:

tan(θ) = opposite / adjacent = cot(φ) / 1 = cot(φ)

However, since we are dealing with the angle made by the dip needle, we need to adjust our understanding. The relationship can be expressed as:

tan(θ) = cot(φ) = 1/tan(φ)

Thus, we can express the angle of dip in terms of the cotangent function:

θ = tan^-1(cot(φ))

Choosing the Correct Option

Now, looking at the options provided:

• 1) tan^-1(cot cos φ)
• 2) tan^-1(tan cos φ)
• 3) tan^-1(cot sec φ)
• 4) tan^-1(tan sin φ)

From our analysis, the correct expression for the angle of dip θ in terms of φ is:

θ = tan^-1(cot φ)

Thus, the answer is option 3: tan^-1(cot sec φ).

Conclusion

In summary, the angle of dip can be derived from the orientation of the dip needle in relation to the magnetic meridian and the horizontal plane. By applying trigonometric identities, we can find that the angle of dip is represented by the cotangent function, leading us to the correct answer among the options provided.