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Chapter 22: Problem 49

ssm The earth's magnetic field, like any magnetic field, stores energy. The maximum strength of the earth's field is about \(7.0 \times 10^{-5} \mathrm{T}\) . Find the maximum magnetic energy stored in the space above a city if the space occupies an area of \(5.0 \times 10^{8} \mathrm{m}^{2}\) and has a height of 1500 \(\mathrm{m}\) .

Short Answer

Expert verified
The maximum magnetic energy stored is approximately 1.46 脳 10鹿鈦� J.

Step by step solution

01

Calculate the volume of the space above the city

To find the volume of space above the city that contains the magnetic field, we multiply the area by the height. \[\text{Volume} = \text{Area} \times \text{Height} = (5.0 \times 10^{8} \, \mathrm{m}^{2}) \times 1500 \, \mathrm{m} \]\[\text{Volume} = 7.5 \times 10^{11} \, \mathrm{m}^{3}\]
02

Use the formula for magnetic energy density

The energy density \(u\) of a magnetic field can be calculated using the formula:\[u = \frac{B^2}{2\mu_0}\]where \(B = 7.0 \times 10^{-5} \, \mathrm{T}\) is the magnetic field strength and \(\mu_0 = 4\pi \times 10^{-7} \, \mathrm{T}\cdot \mathrm{m/A}\) is the permeability of free space.
03

Substitute values into the energy density formula

Substitute the given values into the energy density formula:\[u = \frac{(7.0 \times 10^{-5})^2}{2 \times 4\pi \times 10^{-7}}\]Calculate the expression within the fraction.
04

Calculate the magnetic energy density

After computing:\[u = \frac{(4.9 \times 10^{-9})}{2 \times 4\pi \times 10^{-7}} = \frac{4.9 \times 10^{-9}}{8\pi \times 10^{-7}}\]\[u \approx 1.95 \times 10^{-2} \, \mathrm{J/m}^{3}\]
05

Calculate the total energy stored

The total energy \(U\) stored in the magnetic field is found by multiplying the energy density by the volume:\[U = u \times \text{Volume} = 1.95 \times 10^{-2} \, \mathrm{J/m}^{3} \times 7.5 \times 10^{11} \, \mathrm{m}^{3}\]\[U \approx 1.46 \times 10^{10} \, \mathrm{J}\]

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Key Concepts

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

Earth's Magnetic Field
The Earth's magnetic field is an invisible force field that surrounds our planet. It is generated by the movement of molten iron within Earth's core, acting like a giant bar magnet. This magnetic field is vital as it protects Earth from harmful cosmic radiation and solar wind particles. Over time, the magnetic poles of Earth have shifted, and they can even reverse, though this occurs over thousands of years. Here are some important facts about Earth's magnetic field:
  • It extends tens of thousands of kilometers into space, forming the magnetosphere.
  • The strength of the magnetic field varies across different parts of the world, being strongest at the poles and weakest at the equator.
  • The average field strength at the Earth's surface is around 25 to 65 microteslas.
The maximum strength of the field above a specific area is crucial when calculating aspects like magnetic energy, as seen in exercises requiring such analysis. Understanding the properties of Earth's magnetic field helps us appreciate its essential role in navigation and technological systems.
Magnetic Energy Density
Magnetic energy density is a measure of how much energy is stored in a magnetic field within a given volume. It is vital for understanding how magnetic fields interact with the environment and materials. The formula to calculate magnetic energy density is given by:\[u = \frac{B^2}{2\mu_0}\]where:
  • \(B\) is the magnetic field strength
  • \(\mu_0\) is the permeability of free space
This formula shows that the energy density is proportional to the square of the field strength. Calculating the energy density allows us to determine the total energy stored in a magnetic field when considering the volume it occupies. For practical applications such as in the original exercise, calculating the energy density provides insights into how much magnetic energy is stocked in specific environments, which can be crucial for understanding its impact on surrounding regions.
Permeability of Free Space
The permeability of free space, often denoted as \(\mu_0\), is a constant that characterizes the ability of a vacuum to support a magnetic field. It influences the relationship between magnetic field strength and magnetic flux, playing a fundamental role in electromagnetism. The value of \(\mu_0\) is approximately:\[\mu_0 = 4\pi \times 10^{-7} \, \mathrm{T}\cdot \mathrm{m/A}\]This constant helps define how magnetic fields interact with the environment and materials. It appears in various equations relating to magnetism, most notably in Ampere's law and the equation for magnetic energy density.Understanding permeability is essential because:
  • It aids in deriving important electromagnetic relationships.
  • It allows accurate predictions of how magnetic fields will behave in different settings, such as in engineering and physics applications.
  • It ensures that calculations involving magnetic fields, like those of energy storage as seen in the exercise, are precise and meaningful.
By incorporating \(\mu_0\), calculations can accurately reflect real-world scenarios and contribute to advancements in technology and science.

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Most popular questions from this chapter

Two coils of wire are placed close together. Initially, a current of 2.5 \(\mathrm{A}\) exists in one of the coils, but there is no current in the other. The current is then switched off in a time of \(3.7 \times 10^{-2}\) s. During this time, the average emf induced in the other coil is 1.7 \(\mathrm{V}\) . What is the mutual inductance of the two-coil system?

ssm A \(3.0-\mu \mathrm{F}\) capacitor has a voltage of 35 \(\mathrm{V}\) between its plates. What must be the current in a \(5.0-\mathrm{mH}\) inductor so that the energy stored in the inductor equals the energy stored in the capacitor?

A conducting coil of 1850 turns is connected to a galvanometer, and the total resistance of the circuit is \(45.0 \Omega .\) The area of each turn is \(4.70 \times 10^{-4} \mathrm{~m}^{2} .\) This coil is moved from a region where the magnetic field is zero into a region where it is nonzero, the normal to the coil being kept parallel to the magnetic field. The amount of charge that is induced to flow around the circuit is measured to be \(8.87 \times 10^{-3} \mathrm{C}\). Find the magnitude of the magnetic field.

A copper rod is sliding on two conducting rails that make an angle of 19 with respect to each other, as in the drawing. The rod is moving to the right with a constant speed of 0.60 \(\mathrm{m} / \mathrm{s} .\) A \(0.38-\mathrm{T}\) uniform magnetic field is perpendicular to the plane of the paper. Determine the magnitude of the average emf induced in the triangle \(A B C\) during the 6.0 -s period after the rod has passed point \(A\) .

A square loop of wire consisting of a single turn is perpendicular to a uniform magnetic field. The square loop is then re-formed into a circular loop, which also consists of a single turn and is also perpendicular to the same magnetic field. The magnetic flux that passes- through the square loop is \(7.0 \times 10^{-3}\) Wb. What is the flux that passes through the circular loop?
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