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Chapter 32: Problem 33

On a clear day at a certain location, a \(100-\mathrm{V} / \mathrm{m}\) vertical electric field exists near the Earth's surface. At the same place, the Earth's magnetic field has a magnitude of \(0.500 \times 10^{-4} \mathrm{T} .\) Compute the energy densities of the two fields.

Short Answer

Expert verified
After performing the calculations you will get the energy density for the electric field to be approximately \(4.43 \times 10^{-8}\) J/m鲁 and for the magnet field approximately \(0.99 \times 10^{-8}\) J/m鲁. Note: The answer may vary depending on the accuracy of the calculations.

Step by step solution

01

Compute the Electric Energy Density

The energy density (\(u_E\)) of an electric field is given by the formula: \(u_E = 0.5 \, \epsilon_0 \, E^2\), where \(\epsilon_0 = 8.85 \times 10^{-12} \, \mathrm{C^2/N \cdot m^2}\) is the permittivity of free space and \(E\) is the electric field. In this case, \(E = 100 \, \mathrm{V/m}\), therefore we can substitute these values into the formula: \(u_E = 0.5 \times 8.85 \times 10^{-12} \, \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2 \times (100 \, \mathrm{V/m})^2\).
02

Solve for Electric Energy Density

Calculate the expression got in Step 1 to get the value for \(u_E\).
03

Compute the Magnetic Energy Density

The energy density (\(u_B\)) of a magnetic field is given by the formula: \(u_B = \frac{B^2}{2\mu_0}\), where \(B\) is the magnetic field and \(\mu_0 = 4\pi \times 10^{-7} \, \mathrm{T \cdot m/A}\) is the permeability of free space. The given magnetic field \(B = 0.500 \times 10^{-4} \, \mathrm{T}\), so we substitute these values into the formula: \(u_B = \frac{(0.500 \times 10^{-4} \, \mathrm{T})^2}{2 \times 4\pi \times 10^{-7} \, \mathrm{T \cdot m/A}}\).
04

Solve for Magnetic Energy Density

Calculate the expression got in Step 3 to get the value for \(u_B\).

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Key Concepts

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

Electric Field
An electric field is a field around a charged particle where other charged particles experience a force. Imagine holding a balloon with static electricity. The space around the balloon forms an electric field. In this field, any nearby charges feel a push or pull. The electric field strength, represented by \(E\), is measured in volts per meter (\(\mathrm{V/m}\)). It tells us how strong the force is at each point in space.

To calculate the energy density of an electric field, we use the formula: \(u_E = 0.5 \epsilon_0 E^2\). Here, \(\epsilon_0\) is the permittivity of free space. Understanding this allows you to see how much energy is stored in a given volume of the electric field.
Magnetic Field
Just like electric fields surround charges, magnetic fields surround magnets or current-carrying wires. If you've ever played with magnets, you've felt this invisible push and pull. The magnetic field strength, identified as \(B\), is measured in teslas (\(\mathrm{T}\)). This value can tell us how the magnetic field behaves and interacts with materials.

The energy density of a magnetic field can be calculated using: \(u_B = \frac{B^2}{2\mu_0}\). In this formula, \(\mu_0\) is the permeability of free space. Knowing this helps us understand how energy is distributed within the magnetic field.
Permittivity of Free Space
Permittivity of free space, or \(\epsilon_0\), is a fundamental constant that tells us how electric fields behave in a vacuum. Its value is approximately \(8.85 \times 10^{-12} \, \mathrm{C^2/N \cdot m^2}\). This essentially means it shows how much electric field is "permitted" to exist in free space.

Permittivity is a key part of the electric energy density equation, \(u_E = 0.5 \epsilon_0 E^2\). Its role is to scale the electric field strength to find out how much energy is packed in that field. Think of it as an indicator of how easily electric fields form in a vacuum.
Permeability of Free Space
Permeability of free space, denoted as \(\mu_0\), is another fundamental constant that indicates how magnetic fields interact in a vacuum. Its value stands at \(4\pi \times 10^{-7} \, \mathrm{T \cdot m/A}\). Essentially, it measures how well a space supports magnetic fields.

In the formula for magnetic energy density, \(u_B = \frac{B^2}{2\mu_0}\), the permeability plays a vital role. It helps calculate the amount of energy held within a magnetic field. It's crucial for understanding how magnetic fields move and change in free space.

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Most popular questions from this chapter

Superconducting power transmission. The use of superconductors has been proposed for power transmission lines. A single coaxial cable (Fig. P32.78) could carry \(1.00 \times 10^{3} \mathrm{MW}\) (the output of a large power plant) at \(200 \mathrm{kV},\) DC, over a distance of \(1000 \mathrm{km}\) without loss. An inner wire of radius \(2.00 \mathrm{cm},\) made from the superconductor \(\mathrm{Nb}_{3} \mathrm{Sn},\) carries the current \(I\) in one direction. A surrounding superconducting cylinder, of radius \(5.00 \mathrm{cm},\) would carry the return current \(I\) In such a system, what is the magnetic field (a) at the surface of the inner conductor and (b) at the inner surface of the outer conductor? (c) How much energy would be stored in the space between the conductors in a \(1000-\mathrm{km}\) superconducting line? (d) What is the pressure exerted on the outer conductor?

The resistance of a superconductor. In an experiment carried out by S. C. Collins between 1955 and \(1958,\) a current was maintained in a superconducting lead ring for \(2.50 \mathrm{yr}\) with no observed loss. If the inductance of the ring was \(3.14 \times 10^{-8} \mathrm{H},\) and the sensitivity of the experiment was 1 part in \(10^{9}\), what was the maximum resistance of the ring? (Suggestion: Treat this as a decaying current in an \(R L\) circuit, and recall that \(e^{-x} \approx 1-x\) for small \(x\).)

A self-induced emf in a solenoid of inductance \(L\) changes in time as \(\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{0} e^{-k t} .\) Find the total charge that passes through the solenoid, assuming the charge is finite.

A capacitor in a series \(L C\) circuit has an initial charge \(Q\) and is being discharged. Find, in terms of \(L\) and \(C\), the flux through each of the \(N\) turns in the coil, when the charge on the capacitor is \(Q / 2.\)

Two coils are close to each other. The first coil carries a timevarying current given by \(I(t)=(5.00 \mathrm{A}) e^{-0.0250 t} \sin (377 t)\) At \(t=0.800 \mathrm{s},\) the emf measured across the second coil is \(-3.20 \mathrm{V} .\) What is the mutual inductance of the coils?
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