# 16.3 Energy carried by electromagnetic waves  (Page 4/5)

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How does the intensity of an electromagnetic wave depend on its electric field? How does it depend on its magnetic field?

What is the physical significance of the Poynting vector?

It has the magnitude of the energy flux and points in the direction of wave propagation. It gives the direction of energy flow and the amount of energy per area transported per second.

A 2.0-mW helium-neon laser transmits a continuous beam of red light of cross-sectional area $0.25\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}$ . If the beam does not diverge appreciably, how would its rms electric field vary with distance from the laser? Explain.

## Problems

While outdoors on a sunny day, a student holds a large convex lens of radius 4.0 cm above a sheet of paper to produce a bright spot on the paper that is 1.0 cm in radius, rather than a sharp focus. By what factor is the electric field in the bright spot of light related to the electric field of sunlight leaving the side of the lens facing the paper?

A plane electromagnetic wave travels northward. At one instant, its electric field has a magnitude of 6.0 V/m and points eastward. What are the magnitude and direction of the magnetic field at this instant?

The magnetic field is downward, and it has magnitude $2.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}\text{T}$ .

The electric field of an electromagnetic wave is given by
$E=\left(6.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{V/m}\right)\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\left[2\pi \left(\frac{x}{18\phantom{\rule{0.2em}{0ex}}\text{m}}-\frac{t}{6.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}\phantom{\rule{0.2em}{0ex}}\text{s}}\right)\right]\stackrel{^}{j}.$
Write the equations for the associated magnetic field and Poynting vector.

A radio station broadcasts at a frequency of 760 kHz. At a receiver some distance from the antenna, the maximum magnetic field of the electromagnetic wave detected is $2.15\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-11}\text{T}$ .
(a) What is the maximum electric field? (b) What is the wavelength of the electromagnetic wave?

a. $6.45\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{V/m;}$ b. 394 m

The filament in a clear incandescent light bulb radiates visible light at a power of 5.00 W. Model the glass part of the bulb as a sphere of radius ${r}_{0}=3.00\phantom{\rule{0.2em}{0ex}}\text{cm}$ and calculate the amount of electromagnetic energy from visible light inside the bulb.

At what distance does a 100-W lightbulb produce the same intensity of light as a 75-W lightbulb produces 10 m away? (Assume both have the same efficiency for converting electrical energy in the circuit into emitted electromagnetic energy.)

11.5 m

An incandescent light bulb emits only 2.6 W of its power as visible light. What is the rms electric field of the emitted light at a distance of 3.0 m from the bulb?

A 150-W lightbulb emits 5% of its energy as electromagnetic radiation. What is the magnitude of the average Poynting vector 10 m from the bulb?

$5.97\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}{\phantom{\rule{0.2em}{0ex}}\text{W/m}}^{2}$

A small helium-neon laser has a power output of 2.5 mW. What is the electromagnetic energy in a 1.0-m length of the beam?

At the top of Earth’s atmosphere, the time-averaged Poynting vector associated with sunlight has a magnitude of about $1.4\phantom{\rule{0.2em}{0ex}}{\text{kW/m}}^{2}.$
(a) What are the maximum values of the electric and magnetic fields for a wave of this intensity? (b) What is the total power radiated by the sun? Assume that the Earth is $1.5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{11}\text{m}$ from the Sun and that sunlight is composed of electromagnetic plane waves.

$\text{a.}\phantom{\rule{0.2em}{0ex}}{E}_{0}=1027\phantom{\rule{0.2em}{0ex}}\text{V/m},{B}_{0}=3.42\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\text{T};\phantom{\rule{0.2em}{0ex}}\text{b.}\phantom{\rule{0.2em}{0ex}}3.96\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{26}\phantom{\rule{0.2em}{0ex}}\text{W}$

The magnetic field of a plane electromagnetic wave moving along the z axis is given by $\stackrel{\to }{B}={B}_{0}\left(\text{cos}\phantom{\rule{0.2em}{0ex}}kz+\omega t\right)\stackrel{^}{j}$ , where ${B}_{0}=5.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-10}\phantom{\rule{0.2em}{0ex}}\text{T}$ and $k=3.14\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{-1}.$
(a) Write an expression for the electric field associated with the wave. (b) What are the frequency and the wavelength of the wave? (c) What is its average Poynting vector?

What is the intensity of an electromagnetic wave with a peak electric field strength of 125 V/m?

$20.8\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$

Assume the helium-neon lasers commonly used in student physics laboratories have power outputs of 0.500 mW. (a) If such a laser beam is projected onto a circular spot 1.00 mm in diameter, what is its intensity? (b) Find the peak magnetic field strength. (c) Find the peak electric field strength.

An AM radio transmitter broadcasts 50.0 kW of power uniformly in all directions. (a) Assuming all of the radio waves that strike the ground are completely absorbed, and that there is no absorption by the atmosphere or other objects, what is the intensity 30.0 km away? ( Hint: Half the power will be spread over the area of a hemisphere.) (b) What is the maximum electric field strength at this distance?

a. $4.42\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{‒6}{\phantom{\rule{0.2em}{0ex}}\text{W/m}}^{2}$ ; b. $5.77\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{‒2}\phantom{\rule{0.2em}{0ex}}\text{V/m}$

Suppose the maximum safe intensity of microwaves for human exposure is taken to be $1.00{\phantom{\rule{0.2em}{0ex}}\text{W/m}}^{2}$ . (a) If a radar unit leaks 10.0 W of microwaves (other than those sent by its antenna) uniformly in all directions, how far away must you be to be exposed to an intensity considered to be safe? Assume that the power spreads uniformly over the area of a sphere with no complications from absorption or reflection. (b) What is the maximum electric field strength at the safe intensity? (Note that early radar units leaked more than modern ones do. This caused identifiable health problems, such as cataracts, for people who worked near them.)

A 2.50-m-diameter university communications satellite dish receives TV signals that have a maximum electric field strength (for one channel) of $7.50\phantom{\rule{0.2em}{0ex}}\text{μV/m}$ (see below). (a) What is the intensity of this wave? (b) What is the power received by the antenna? (c) If the orbiting satellite broadcasts uniformly over an area of $1.50\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{13}{\phantom{\rule{0.2em}{0ex}}\text{m}}^{2}$ (a large fraction of North America), how much power does it radiate? a. $7.47\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-14}{\phantom{\rule{0.2em}{0ex}}\text{W/m}}^{2}$ ; b. $3.66\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{\text{−}13}\phantom{\rule{0.2em}{0ex}}\text{W}$ ; c. 1.12 W

Lasers can be constructed that produce an extremely high intensity electromagnetic wave for a brief time—called pulsed lasers. They are used to initiate nuclear fusion, for example. Such a laser may produce an electromagnetic wave with a maximum electric field strength of $1.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{11}\phantom{\rule{0.2em}{0ex}}\text{V}\text{/}\text{m}$ for a time of 1.00 ns. (a) What is the maximum magnetic field strength in the wave? (b) What is the intensity of the beam? (c) What energy does it deliver on an $1.00{\text{-mm}}^{2}$ area?

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