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Chapter 22: Problem 45

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 \(1.46025 \times 10^{10} \mathrm{~J}\).

Step by step solution

01

Identify Given Values

We are given the maximum magnetic field strength \(B = 7.0 \times 10^{-5} \mathrm{~T}\), the area \(A = 5.0 \times 10^{8} \mathrm{~m}^{2}\), and the height \(h = 1500 \mathrm{~m}\). These represent the important variables in the problem: the magnetic field strength, the cross-sectional area through which the field extends, and the volume height.
02

Calculate the Volume Above the City

To find the volume \(V\) occupied by the space above the city, use the formula for the volume of a rectangular prism: \(V = A \times h\). Substituting in the given values, we have \(V = 5.0 \times 10^{8} \mathrm{~m}^{2} \times 1500 \mathrm{~m}\), which gives us \(V = 7.5 \times 10^{11} \mathrm{~m}^{3}\).
03

Determine the Magnetic Energy Density

The energy density \(u\) stored in a magnetic field is given by the formula \(u = \frac{B^2}{2\mu_0}\), where \(\mu_0 = 4\pi \times 10^{-7} \mathrm{~T}\cdot\mathrm{m/A}\) is the permeability of free space. Substituting in \(B = 7.0 \times 10^{-5} \mathrm{~T}\), we find \(u = \frac{(7.0 \times 10^{-5})^2}{2 \times 4\pi \times 10^{-7}}\). This simplifies to \(u = \frac{4.9 \times 10^{-9}}{8\pi \times 10^{-7}} \approx 1.947 \times 10^{-2} \mathrm{~J/m}^3\).
04

Calculate the Magnetic Energy Stored

The total magnetic energy \(E\) stored is the product of the energy density \(u\) and the volume \(V\): \(E = u \times V\). Substituting in the value of \(u = 1.947 \times 10^{-2} \mathrm{~J/m^3}\) and \(V = 7.5 \times 10^{11} \mathrm{~m}^{3}\), we get \(E \approx 1.947 \times 10^{-2} \times 7.5 \times 10^{11} \approx 1.46025 \times 10^{10} \mathrm{~J}\).

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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 a fascinating natural phenomenon that surrounds our planet. It is akin to a giant magnet buried inside the Earth, with its magnetic field lines extending far into space. The field is strongest near the poles and weakest near the equator. This magnetic field is vital for life on Earth as it deflects harmful solar wind particles, acting like a protective shield. Understanding this field can help us comprehend how certain navigational tools, like compasses, work since they align themselves along these magnetic lines. Another interesting fact is that the magnetic field varies in strength across different regions. The maximum strength mentioned in this exercise, around \(7.0 \times 10^{-5} \mathrm{~T}\), reveals just how subtle yet significant these magnetic forces can be. By tapping into this unseen force, we can explore the immense potential of the natural world.
Energy Density in Magnetic Fields
Energy density refers to the amount of energy stored in a given volume of space. When discussing magnetic fields, it's crucial to know that energy density quantifies how much energy is packed into this field. The formula used is \(u = \frac{B^2}{2\mu_0}\), where \(B\) is the magnetic field strength, and \(\mu_0\) is the permeability of free space. Permeability helps understand how a magnetic field permeates through space or a material. Essentially, energy density lets us measure how powerful the magnetic field is in terms of energy.
  • The stronger the magnetic field \(B\), the higher the energy density \(u\).
  • The relationship highlights that even a small increase in \(B\) results in a significant increase in \(u\), due to the square of \(B\) in the formula.
Analyzing energy density helps scientists and engineers evaluate a magnetic field's potential to do work, such as in electrical devices and magnetic energy storage systems. The calculation shown in the original exercise provides insight as to how much magnetic energy is contained within a particular volume of space.
Magnetic Field Strength
Magnetic field strength is a measure of how intense a magnetic field is at a given point. It is denoted by the symbol \(B\) and is measured in teslas (\(\mathrm{T}\)). In the context of the Earth's magnetic field, this strength determines how powerfully the magnetic field is exerting its influence over an area. For most everyday purposes,
  • A magnetic field strength of several microteslas (\(\mathrm{\mu T}\)) is what a typical compass interacts with.
  • Strong magnetic fields can be generated in laboratories or using electromagnets, reaching strengths far beyond natural levels.
Understanding magnetic field strength is important not just for practical navigation but also for researching the Earth's interior and for applications in various technologies. In this exercise, the given field strength \(7.0 \times 10^{-5} \mathrm{~T}\) indicates a relatively weak field compared to some other sources, yet its power is undeniable in the context of global magnetic interactions.

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Most popular questions from this chapter

Interactive Solution \(\underline{22.39}\) at provides one model for solving this problem. The maximum strength of the earth's magnetic field is about \(6.9 \times 10^{-5} \mathrm{~T}\) near the south magnetic pole. In principle, this field could be used with a rotating coil to generate 60.0 Hz ac electricity. What is the minimum number of turns (area per turn \(=0.022 \mathrm{~m}^{2}\) ) that the coil must have to produce an rms voltage of \(120 \mathrm{~V} ?\)

The coil within an ac generator has an area per turn of \(1.2 \times 10^{-2} \mathrm{~m}^{2}\) and consists of 500 turns. The coil is situated in a 0.13-T magnetic field and is rotating at an angular speed of \(34 \mathrm{rad} / \mathrm{s}\). What is the emf induced in the coil at the instant when the normal to the loop makes an angle of \(27^{\circ}\) with respect to the direction of the magnetic field?

A copper rod is sliding on two conducting rails that make an angle of \(19^{\circ}\) 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 circular coil \((950\) turns, radius \(=0.060 \mathrm{~m})\) is rotating in a uniform magnetic field. At \(t=0 \mathrm{~s}\), the normal to the coil is per pendicular to the magnetic field. At \(t=0.010 \mathrm{~s}\) the normal makes an angle of \(\phi=45^{\circ}\) with the field because the coil has made oneeighth of a revolution. An average emf of magnitude \(0.065 \mathrm{~V}\) is induced in the coil. Find the magnitude of the magnetic field at the location of the coil.

A vacuum cleaner is plugged into a \(120.0\) -V socket and uses \(3.0\) A of current in normal operation when the back emf generated by the electric motor is \(72.0 \mathrm{~V}\). Find the coil resistance of the motor.
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