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Chapter 20: Problem 77

How much energy due to Earth's magnetic field is present in $1.0 \mathrm{m}^{3}$ of space near Earth's surface, at a place where the field has magnitude \(0.45 \mathrm{mT} ?\)

Short Answer

Expert verified
Answer: The energy is approximately 1.91 × 10⁻⁷ J.

Step by step solution

01

Calculate the energy density (u) in the magnetic field

Using the formula for the energy density in a magnetic field, we can plug in the given value for the magnetic field magnitude and the permeability of free space. \(B = 0.45 \times 10^{-3} \mathrm{T}\) (converting mT to T) \(u = \frac{1}{2(4\pi\times10^{-7})}(0.45\times10^{-3})^2\)
02

Calculate the energy density (u) value

Solving for u: \(u \approx 1.91\times10^{-7}\frac{\mathrm{J}}{{m}^{3}}\) Now, we have the energy density in the magnetic field.
03

Calculate the energy present in the volume of the space

We can multiply the energy density (u) by the given volume to find the total energy present in this volume of space. \(E = u \cdot V \) \(E = (1.91\times10^{-7}\mathrm{\frac{J}}{{m}^{3}}) \cdot (1.0\mathrm{m}^{3})\)
04

Calculate the energy (E) value

Solving for E: \(E \approx 1.91\times10^{-7}\mathrm{J}\) Therefore, the energy due to Earth's magnetic field present in \(1.0\mathrm{m}^{3}\) of space near Earth's surface, at a place where the field has a magnitude \(0.45\mathrm{mT}\), is approximately \(1.91\times10^{-7}\mathrm{J}\).

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Most popular questions from this chapter

A de motor is connected to a constant emf of \(12.0 \mathrm{V}\) The resistance of its windings is \(2.0 \Omega .\) At normal operating speed, the motor delivers \(6.0 \mathrm{W}\) of mechanical power. (a) What is the initial current drawn by the motor when it is first started up? (b) What current does it draw at normal operating speed? (c) What is the back emf induced in the windings at normal speed?
A transformer with a primary coil of 1000 turns is used to step up the standard \(170-\mathrm{V}\) amplitude line voltage to a \(220-\mathrm{V}\) amplitude. How many turns are required in the secondary coil?
A flip coil is a device used to measure a magnetic field. A coil of radius $r, N\( turns, and electrical resistance \)R$ is initially perpendicular to a magnetic field of magnitude B. The coil is connected to a special kind of galvanometer that measures the total charge \(Q\) that flows through it. To measure the field, the flip coil is rapidly flipped upside down. (a) What is the change in magnetic flux through the coil in one flip? (b) If the time interval during which the coil is flipped is \(\Delta t,\) what is the average induced emf in the coil? (c) What is the average current that flows through the galvanometer? (d) What is the total charge \(Q\) in terms of \(r, N, R,\) and \(B ?\)
A transformer for an answering machine takes an ac voltage of amplitude $170 \mathrm{V}\( as its input and supplies a \)7.8-\mathrm{V}$ amplitude to the answering machine. The primary has 300 turns. (a) How many turns does the secondary have? (b) When idle, the answering machine uses a maximum power of \(5.0 \mathrm{W}\). What is the amplitude of the current drawn from the 170 -V line?
An airplane is flying due north at \(180 \mathrm{m} / \mathrm{s}\). Earth's magnetic field has a northward component of \(0.30 \mathrm{mT}\) and an upward component of \(0.38 \mathrm{mT}\). (a) If the wingspan (distance between the wingtips) is \(46 \mathrm{m},\) what is the motional emf between the wingtips? (b) Which wingtip is positively charged?
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