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Chapter 32: Problem 9

In a certain region the average horizontal component of Earth鈥檚 magnetic field in 1912 was 14 mT, and the average inclination or 鈥渄ip鈥� was 70掳. What was the corresponding magnitude of Earth鈥檚 magnetic field?

Short Answer

Expert verified
The magnitude of Earth's magnetic field is approximately 40.94 mT.

Step by step solution

01

Understand the Problem Statement

We are given the horizontal component and the inclination (or dip) angle of Earth's magnetic field. We need to find the total magnitude of Earth's magnetic field using this information.
02

Understand the Relationship Between Components

The Earth's magnetic field can be resolved into horizontal (\(B_H\)) and vertical (\(B_V\)) components. The total magnitude (\(B\)) can be determined using the Pythagorean theorem as follows:\[ B = \sqrt{B_H^2 + B_V^2} \] where\(B_V = B \sin(\theta)\)and\(B_H = B \cos(\theta)\).
03

Calculate the Vertical Component

Given the inclination angle (\(\theta = 70^\circ\)), we can express the vertical component in terms of the total magnitude:\[ B_V = B \sin(70^\circ) \] Use the relationship between horizontal and vertical components. Since \(B_H = B \cos(\theta)\),we plug in the given horizontal component (14 mT):\[ 14 = B \cos(70^\circ) \].
04

Solve for the Magnitude of Earth's Magnetic Field

Rearrange the equation from Step 3:\[ B = \frac{14}{\cos(70^\circ)} \]Calculate \(\cos(70^\circ) \approx 0.3420\):\[ B = \frac{14}{0.3420} \approx 40.94 \, \text{mT} \].
05

Verify the Result

Verify that the calculated magnitude satisfies both component equations. Plug the value back into the \(B_H\) equation:\( B_H = 40.94 \times 0.3420 \approx 14 \, \text{mT} \),which matches the given horizontal component.Therefore, the solution is consistent and correct.

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Key Concepts

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

Magnetic Field Components
In physics, when we consider Earth's magnetic field, it is often useful to break it down into different components. This allows us to analyze its behavior more effectively.
For Earth's magnetic field, two main components are considered:
  • **Horizontal Component (\(B_H\))**: This part runs parallel to the surface of the Earth and is key in navigation, such as guiding compasses.
  • **Vertical Component (\(B_V\))**: This runs perpendicular, pointing into or out of the Earth, and is associated with the magnetic "dip" or "inclination" angle.
By understanding these elements, we can use mathematical formulas to find the total magnetic field and its behavior in a specific region.
Inclination Angle
The inclination angle, also known as magnetic dip, is the angle made between the total magnetic field direction and the horizontal plane.
This angle is crucial because it describes how the magnetic field lines plunge into the Earth鈥檚 surface. Generally:
  • At the magnetic poles, this angle is close to 90掳, because the lines are nearly vertical.
  • Near the magnetic equator, the inclination is about 0掳, as the lines are horizontal.
In the problem, the dip angle is 70掳, which means the field lines are steeply inclined but not completely vertical.
Understanding the inclination provides insight into the strength and direction of Earth鈥檚 magnetic field at a specific location.
Pythagorean Theorem
The Pythagorean theorem is a powerful mathematical tool used to connect the components of Earth's magnetic field to its total magnitude.
It states that for a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides. In our context, the theorem is used as follows:\[B = \sqrt{B_H^2 + B_V^2}\]Here, the total magnetic field \(B\)is the hypotenuse, while \(B_H\)and \(B_V\)are the perpendicular components. To solve for \(B\): simply square the horizontal and vertical components, add them, and take the square root.
This mathematical principle simplifies the analysis of Earth's magnetic field from its components.
Magnetic Dip
The magnetic dip is essentially synonymous with the inclination angle.
It is defined as the angle at which the Earth's magnetic field lines intersect the plane of the horizontal component. Understanding this dip helps scientists measure and understand the Earth鈥檚 magnetism at different locations. Key points about magnetic dip:
  • It provides information on the tilt of the magnetic field lines.
  • Changes in dip contribute to understanding magnetic anomalies and alignments.
  • Measured by instruments called dip circles, a higher dip indicates a strong vertical component.
Overall, the dip gives vital data about how much of Earth's magnetic field is buried under the surface or pointing towards it.
Physics Problem Solving
Physics problem solving, especially concerning Earth's magnetic field, involves careful analysis and application of theoretical concepts to real-world problems.
The problem-solving process typically includes:
  • **Understanding the Given Information**: Clearly know what is provided, like the horizontal component and inclination angle.
  • **Breaking Down the Problem**: Recognize the relationship between the components of the magnetic field.
  • **Using the Right Formulas**: Apply mathematical theorems like the Pythagorean theorem and trigonometry.
  • **Verifying Results**: Confirm that calculations make sense with the original problem statement.
This structured approach helps avoid errors and ensures the solution accurately reflects real-world physics, similarly to solving the given exercise by connecting all the components methodically.

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Most popular questions from this chapter

The saturation magnetization \(M_{\max }\) of the ferromagnetic metal nickel is \(4.70 \times 10^{5} \mathrm{~A} / \mathrm{m}\). Calculate the magnetic dipole moment of a single nickel atom. (The density of nickel is \(8.90 \mathrm{~g} / \mathrm{cm}^{3}\), and its molar mass is \(58.71 \mathrm{~g} / \mathrm{mol}\).)

A parallel-plate capacitor with circular plates of radius \(40 \mathrm{~mm}\) is being discharged by a current of \(6.0 \mathrm{~A}\). At what radius (a) inside and (b) outside the capacitor gap is the magnitude of the induced magnetic field equal to \(60 \%\) of its maximum value? (c) What is that maximum value?

The induced magnetic field at radial distance \(6.0 \mathrm{~mm}\) from the central axis of a circular parallel-plate capacitor is \(1.2 \times 10^{-7} \mathrm{~T}\). The plates have radius \(4.0 \mathrm{~mm}\). At what rate \(d \vec{E} / d t\) is the electric field between the plates changing?

The exchange coupling mentioned in Module \(32-8\) as being responsible for ferro- magnetism is not the mutual magnetic intermagnetism is not the mutual magnetic interaction between two elementary magnetic dipoles. To show this, calculate (a) the magnitude of the magnetic field a distance of \(10 \mathrm{~nm}\) away, along the dipole axis, from an atom with magnetic dipole moment \(1.5 \times 10^{-25} \mathrm{~J} / \mathrm{T}\) (cobalt), and (b) the minimum energy required to turn a second identical dipole end for end in this field. (c) By comparing the latter with the mean translational kinetic energy of \(0.040 \mathrm{eV}\), what can you conclude?

The magnitude of the magnetic dipole moment of Earth is 8.0 1022 J/T. (a) If the origin of this magnetism were a magnetized iron sphere at the center of Earth, what would be its radius? (b) What fraction of the volume of Earth would such a sphere occupy? Assume complete alignment of the dipoles. The density of Earth鈥檚 inner core is 14 g/cm3 . The magnetic dipole moment of an iron atom is 2.1 1023 J/T. (Note: Earth鈥檚 inner core is in fact thought to be in both liquid and solid forms and partly iron, but a permanent magnet as the source of Earth鈥檚 magnetism has been ruled out by several considerations. For one, the temperature is certainly above the Curie point.)
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