Problem 1974 - JEE Physics

Question

The true value of angle of dip at a place is \(60^{\circ}\), the apparent dip in a plane inclined at an angle of \(30^{\circ}\) with magnetic meridian is. (a) \(\tan ^{-1}(1 / 2)\) (b) \(\tan ^{-1} 2\) (c) \(\tan ^{-1}(2 / 3)\) (d) None of these

Step-by-Step Solution

Answer

The apparent dip in a plane inclined at an angle of \(30^{\circ}\) with the magnetic meridian is \(\tan^{-1}(\frac{\sqrt{3}}{2})\).

Step 1: Write down the given values

We have the following values given: True dip (δ) = \(60^{\circ}\) Inclination of the plane (β) = \(30^{\circ}\)

Step 2: Substitute the given values in the formula

We have the formula: \( \tan\alpha = \tan\delta \sin\beta \). Substitute the given values of true dip and inclination of the plane: \(\tan\alpha = \tan(60^{\circ}) \sin(30^{\circ})\)

Step 3: Calculate the value of tanα

Now, calculate the value of \(\tan\alpha\): \( \tan\alpha = \tan(60^{\circ}) \sin(30^{\circ}) = \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \)

Step 4: Find the angle α

To find the angle α, we need to find the inverse tangent of the obtained value: \(\alpha = \tan^{-1}(\frac{\sqrt{3}}{2})\)

Step 5: Choose the correct option

From the given choices, option (c) is \(\tan ^{-1}(2 / 3)\), which is not equal to \(\tan^{-1}(\frac{\sqrt{3}}{2})\). Therefore, the correct answer is option (d) None of these. The apparent dip in a plane inclined at an angle of \(30^{\circ}\) with the magnetic meridian is \(\tan^{-1}(\frac{\sqrt{3}}{2})\).

Key Concepts

Angle of Dip

In the world of magnetism, the angle of dip, sometimes referred to as magnetic inclination, is the angle made by the Earth's magnetic field lines with the horizontal plane at any given point on the Earth's surface. It's a crucial parameter for understanding how Earth's magnetic field interacts with objects on the surface.

The angle of dip varies from place to place due to the structure of the Earth's magnetic field. At the magnetic poles, the angle of dip is almost 90 degrees because the magnetic field lines are nearly vertical. At the magnetic equator, the field lines run parallel to the surface, resulting in a dip of zero.

For practical purposes, the angle of dip helps in navigation and understanding the magnetic environment. Aviation and maritime industries, for example, rely on dip readings for accurate compass calibration.

Apparent Dip

Apparent dip is a concept used to describe what happens when a plane is tilted with respect to the magnetic meridian. Imagine you're in a place where the true dip is known, but the surface you're measuring it on isn't flat or aligned with the magnetic meridian.

When the plane is inclined, the observed angle, called the apparent dip, will differ from the true dip. This occurs because the tilt of the plane changes the interaction with the magnetic field. Calculating apparent dip requires adjustments using trigonometric relationships. The formula used is:

• \( \tan\alpha = \tan\delta \sin\beta \)

where

• \( \alpha \) is the apparent dip,
• \( \delta \) is the true dip,
• \( \beta \) is the angle between the plane and the magnetic meridian.

Understanding apparent dip is essential for geological studies and compass readings when the local terrains are uneven or structures are inclined.

Magnetic Meridian

The term 'magnetic meridian' refers to an imaginary line around Earth, which connects the magnetic north and south poles. Think of it as an invisible vertical plane through which Earth's magnetic field lines travel.

A compass needle aligns itself with the magnetic meridian because it aligns with the Earth's magnetic field. This makes the concept crucial for navigation. Aligning measurements or devices to the magnetic meridian ensures that the magnetic field lines are accurately followed.

In practical applications, magnetic meridians are used to ensure instruments and measurements are correctly orientated. This is essential not only for navigation but also for geological and geophysical surveys, where understanding Earth's magnetic influence is paramount.