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Chapter 34: Problem 4

An electromagnetic wave in vacuum has an electric field amplitude of \(220 \mathrm{V} / \mathrm{m} .\) Calculate the amplitude of the corresponding magnetic field.

Short Answer

Expert verified
The amplitude of the corresponding magnetic field is approximately \(7.3 脳 10^{-7} T\).

Step by step solution

01

Identify given values

Identify the values given in the problem. The amplitude of the electric field, denoted as \(E_0\), is given as 220 volts/meter. The speed of light in vacuum, denoted as \(c\), is a standard value of \(3.0 脳 10^8 m/s\).
02

Write down the initial equation

Write down the equation to calculate the amplitude of the magnetic field, which is \(B_0 = \frac{E_0}{c}\).
03

Substitute the values into the equation

Substitute the given values into the equation. Therefore, we have \(B_0 = \frac{220V/m}{3.0脳10^8m/s}\).
04

Carry out the calculation

Carry out the calculation. The unit 'm' in the numerator and denominator will cancel out, leaving us with an answer in \(Tesla (T)\).

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Key Concepts

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

Electric Field Amplitude
Understanding the electric field amplitude is fundamental when studying electromagnetic waves. The amplitude of an electric field, often denoted as \( E_0 \), represents the maximum strength of the electric field at any point in space. It is measured in volts per meter (V/m) and indicates how much force a charged particle would experience in that field. In practical terms, the larger the amplitude of the electric field, the more energy the electromagnetic wave carries.

As a visualization, imagine a wave on the ocean. The electric field amplitude is analogous to the height of the wave from the calm sea level to the wave's crest. Higher waves mean more energy and, similarly, a higher \( E_0 \) means a stronger electric field鈥攁 greater potential to do work on charged particles. When solving a problem involving electromagnetic waves, determining the electric field amplitude is often the initial step, which then facilitates the calculation of other related quantities, such as the magnetic field amplitude.
Speed of Light in Vacuum
The speed of light in a vacuum, denoted as \( c \), is a fundamental constant of nature that plays a pivotal role in physics, particularly in electromagnetism and relativity. Its exact value is approximately \( 3.0 \times 10^8 \) meters per second. One of the fascinating aspects of this constant is that it is the ultimate speed limit for all matter and information in the universe.

In the context of electromagnetic waves, the speed of light is the speed at which light waves propagate through a vacuum. It is an essential factor when relating the electric field amplitude and the magnetic field amplitude in such waves. The importance of \( c \) in calculations cannot be overstated, as it often serves as a conversion factor between electric and magnetic fields. When dealing with electromagnetic wave problems, the speed of light in a vacuum conveys how quickly changes in the electric field can influence the magnetic field, and vice versa, allowing for the precise synchronization of the oscillations in both fields.
Magnetic Field Amplitude
The magnetic field amplitude in the context of electromagnetic waves corresponds to the maximum value of the magnetic field, which is denoted by \( B_0 \). It is measured in Teslas (T), a unit that quantifies the strength and direction of a magnetic field. In a typical electromagnetic wave problem, such as the one described in the exercise, the magnetic field amplitude is often calculated from the known electric field amplitude and the speed of light in a vacuum.

To put it simply, just as electric field amplitude relates to the force on a charged particle, the magnetic field amplitude relates to the force on a moving charge in a magnetic field. The higher the amplitude \( B_0 \), the stronger the magnetic force that can be exerted on moving charges. The relation between the electric and magnetic fields in an electromagnetic wave 鈥� characterized by the equation \( B_0 = \frac{E_0}{c} \) 鈥� is derived from Maxwell's equations. This equation is instrumental in solving problems involving electromagnetic waves, as it highlights the intrinsic link between the electric and magnetic components of the wave.

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Most popular questions from this chapter

One goal of the Russian space program is to illuminate dark northern cities with sunlight reflected to Earth from a \(200-\mathrm{m}\) diameter mirrored surface in orbit. Several smaller prototypes have already been constructed and put into orbit. (a) Assume that sunlight with intensity \(1340 \mathrm{W} / \mathrm{m}^{2}\) falls on the mirror nearly perpendicularly and that the atmosphere of the Earth allows \(74.6 \%\) of the energy of sunlight to pass through it in clear weather. What is the power received by a city when the space mirror is reflecting light to it? (b) The plan is for the reflected sunlight to cover a circle of diameter \(8.00 \mathrm{km} .\) What is the intensity of light (the average magnitude of the Poynting vector) received by the city? (c) This intensity is what percentage of the vertical component of sunlight at Saint Petersburg in January, when the sun reaches an angle of \(7.00^{\circ}\) above the horizon at noon?

A very large flat sheet carries a uniformly distributed electric current with current per unit width \(J_{s} .\) Example 30.6 demonstrated that the current creates a magnetic field on both sides of the sheet, parallel to the sheet and perpendicular to the current, with magnitude \(B=\frac{1}{2} \mu_{0} J_{s} .\) If the current oscillates in time according to $$ \mathbf{J}_{s}=J_{\max }(\cos \omega t) \hat{\mathbf{j}}=J_{\max }[\cos (-\omega t)] \hat{\mathbf{j}} $$ the sheet radiates an electromagnetic wave as shown in Figure P34.37. The magnetic field of the wave is described by the wave function \(\quad \mathbf{B}=\frac{1}{2} \mu_{0} J_{\max }[\cos (k x-\omega t)] \hat{\mathbf{k}}\) (a) Find the wave function for the electric field in the wave. (b) Find the Poynting vector as a function of \(x\) and \(t\) (c) Find the intensity of the wave. (d) What If? If the sheet is to emit radiation in each direction (normal to the plane of the sheet) with intensity \(570 \mathrm{W} / \mathrm{m}^{2},\) what maximum value of sinusoidal current density is required?

Two hand-held radio transceivers with dipole antennas are separated by a large fixed distance. If the transmitting antenna is vertical, what fraction of the maximum received power will appear in the receiving antenna when it is inclined from the vertical by (a) \(15.0^{\circ} ?\) (b) \(45.0^{\circ} ?\) (c) \(90.0^{\circ} ?\)

A possible means of space flight is to place a perfectly reflecting aluminized sheet into orbit around the Earth and then use the light from the Sun to push this "solar sail." Suppose a sail of area \(6.00 \times 10^{5} \mathrm{m}^{2}\) and mass \(6000 \mathrm{kg}\) is placed in orbit facing the Sun. (a) What force is exerted on the sail? (b) What is the sail's acceleration? (c) How long does it take the sail to reach the Moon, \(3.84 \times 10^{8} \mathrm{m}\) away? Ignore all gravitational effects, assume that the acceleration calculated in part (b) remains constant, and assume a solar intensity of \(1340 \mathrm{W} / \mathrm{m}^{2}\).

Review problem. (a) An elderly couple has a solar water heater installed on the roof of their house (Fig. P34.68). The heater consists of a flat closed box with (IMAGE CANNOT COPY) extraordinarily good thermal insulation. Its interior is painted black, and its front face is made of insulating glass. Assume that its emissivity for visible light is 0.900 and its emissivity for infrared light is \(0.700 .\) Assume that light from the noon Sun is incident perpendicular to the glass with an intensity of \(1000 \mathrm{W} / \mathrm{m}^{2},\) and that no water enters or leaves the box. Find the steady-state temperature of the interior of the box. (b) What If? The couple builds an identical box with no water tubes. It lies flat on the ground in front of the house. They use it as a cold frame, where they plant seeds in early spring. Assuming the same noon Sun is at an elevation angle of \(50.0^{\circ},\) find the steady-state temperature of the interior of this box when its ventilation slots are tightly closed.
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