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Chapter 28: Problem 31

The angle of dip, if a dip needle oscillating in a vertical plane makes 40 oscillations per minute in a magnetic meridian and 30 oscillations per minute in a vertical plane at right angle to the magnetic meridian, is: (a) \(\theta=\sin ^{-1}(0.5625)\) (b) \(\theta=\sin ^{-1}(0.325)\) (c) \(\theta=\sin ^{-1}(0.425)\) (d) \(\theta=\sin ^{-1}(0.235)\)

Short Answer

Expert verified
None of the given options match the calculated value; verify problem accuracy.

Step by step solution

01

Understand the Problem

We need to find the angle of dip using the frequencies of oscillation of a dip needle in two different orientations: in the magnetic meridian and at right angle to it. This involves using the formula for dip angle based on oscillation frequencies.
02

Write Down the Formula

The formula to determine the angle of dip \(\theta\) when given the frequencies of oscillation in the magnetic meridian \(n_1\) and at right angles to it \(n_2\) is: \[ n_1 = n_2 \cos \theta \] Rearranging this formula to solve for \(\cos \theta\), we get: \[ \cos \theta = \frac{n_2}{n_1} \] And subsequently, \(\sin \theta\) will be: \[ \sin \theta = \sqrt{1 - \cos^2 \theta} \]
03

Plug in Given Values

Substitute the given frequencies into the formula. We are given that \(n_1 = 40\) oscillations per minute and \(n_2 = 30\) oscillations per minute. Calculate \(\cos \theta\): \[ \cos \theta = \frac{30}{40} = 0.75 \] Use \(\cos \theta\) to calculate \(\sin \theta\): \[ \sin \theta = \sqrt{1 - (0.75)^2} = \sqrt{1 - 0.5625} = \sqrt{0.4375} \approx 0.661 \]
04

Compare with Options

We know \(\sin \theta \approx 0.661\). Now, let's match this value with the given options: - (a) \(\theta = \sin^{-1}(0.5625)\)- (b) \(\theta = \sin^{-1}(0.325)\)- (c) \(\theta = \sin^{-1}(0.425)\)- (d) \(\theta = \sin^{-1}(0.235)\) Since none of the options provide \(\sin \theta \approx 0.661\), there's been a mistake in providing the list of possible options. Verify that the calculation aligns with the intended result description or check problem criteria.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetic Meridian
The magnetic meridian is an essential concept when understanding the behavior of a dip needle and its oscillations. Imagine a vertical plane that passes through Earth鈥檚 magnetic field lines. This plane, where a magnetic compass needle aligns itself, is known as the magnetic meridian. It connects the magnetic north and south poles, distinguishing itself from the geographical meridian which aligns with true north and south, as defined by Earth's rotational axis.

When you think of the magnetic meridian, consider:
  • It is the imaginary line that follows Earth's magnetic field.
  • Magnetic compasses traditionally align with this meridian.
  • In exercises involving dip needles, oscillations in this plane are key to measuring magnetic properties.
Understanding the magnetic meridian helps to contextualize how dip needles measure the angle of dip based on oscillation frequencies in parallel and perpendicular planes.
Oscillation Frequencies
Oscillation frequencies play a crucial role in understanding how a dip needle determines the angle of dip. The frequency of oscillation refers to how many times the needle swings back and forth in a particular timeframe, such as per minute.

In the context of the exercise, oscillation frequencies are measured in two distinct planes:
  • Within the magnetic meridian.
  • At right angles to the magnetic meridian.
By comparing these frequencies, one can calculate the angle of dip. The relationship between these frequencies and the dip angle is mathematically expressed by:\[ n_1 = n_2\cos \theta\]Here, \(n_1\) is the oscillation frequency in the magnetic meridian and \(n_2\) at right angles to it. Solving for \(\cos \theta\) gives:\[\cos \theta = \frac{n_2}{n_1}\]Calculating these accurately reveals the consistent behavior of a dip needle across different orientations, and ultimately helps in finding the angle.
Dip Needle
The dip needle is a fascinating instrument used to measure the angle of dip. It is essentially a magnetic needle that pivots vertically and can oscillate in a plane, making it valuable for distinguishing magnetic inclinations.

Here's what makes the dip needle especially useful:
  • It provides a direct visual of the magnetic field's direction by aligning with Earth's magnetic lines.
  • When oscillating in the magnetic meridian, it offers insights into Earth's magnetic properties in that direction.
  • Its oscillation frequency, when compared across different planes, helps calculate the angle of dip using trigonometric relationships.
The dip needle simplifies understanding of the 3D orientation of magnetic fields inside Earth, making it a vital tool in geomagnetism and related studies. By carefully measuring how this needle behaves in the magnetic meridian versus perpendicular planes, one can glean valuable data about Earth鈥檚 magnetic inclination.

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Most popular questions from this chapter

The work done in deflecting a small magnet of magnetic moment \(10 \mathrm{Am}^{2}\) through \(180^{\circ}\) from a uniform magnetic field of strength \(0.4 \times 10^{-4} \mathrm{~T}\) is : (a) \(8 \times 10^{-4} \mathrm{~J}\) (b) zero (c) \(4 \times 10^{-4} \mathrm{~J}\) (d) none of these

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A uniform magnetic needle of strength of each pole is \(98.1\) amp. \(\mathrm{cm}\) is suspended from its centre by a thread. When a mass of \(50 \mathrm{mg}\) is loaded to its upper end, the needle become horizontal, then the vertical component of earth's magnetic induction is : \(\left(g=981 \mathrm{~cm} / \mathrm{sec}^{2}\right)\) (a) \(0.50\) gauss (b) \(0.25\) gauss (c) \(0.05\) gauss (d) \(0.005\) gauss

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