Problems & Exercises

Verify that the correct value for the speed of light $c$ is obtained when numerical values for the permeability and permittivity of free space (${\mu }_0$ and ${\epsilon }_0$) are entered into the equation $c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}$.

Show that, when SI units for ${\mu }_0$ and ${\epsilon }_0$ are entered, the units given by the right-hand side of the equation in the problem above are m/s.

The maximum magnetic field strength of an electromagnetic field is $5\times {\text{10}}^{-6}\text{T}$. Calculate the maximum electric field strength if the wave is traveling in a medium in which the speed of the wave is $\text{0.75}c$.

Verify the units obtained for magnetic field strength $B$ in Example 24.1 (using the equation $B=\frac{E}{c}$) are in fact teslas (T).

(a) Calculate the range of wavelengths for AM radio given its frequency range is 540 to 1600 kHz. (b) Do the same for the FM frequency range of 88.0 to 108 MHz.

Find the frequency range of visible light, given that it encompasses wavelengths from 380 to 760 nm.

Electromagnetic radiation having a $\text{15}\text{.}0-\mu \text{m}$ wavelength is classified as infrared radiation. What is its frequency?

A radar used to detect the presence of aircraft receives a pulse that has reflected off an object $6\times {\text{10}}^{-5}\text{s}$ after it was transmitted. What is the distance from the radar station to the reflecting object?

Determine the amount of time it takes for X-rays of frequency $3\times {\text{10}}^{\text{18}}\text{Hz}$ to travel (a) 1 mm and (b) 1 cm.

If the Sun suddenly turned off, we would not know it until its light stopped coming. How long would that be, given that the Sun is $1\text{.}\text{50}\times {\text{10}}^{\text{11}}\text{m}$ away?

Distances in space are often quoted in units of light years, the distance light travels in one year. (a) How many meters is a light year? (b) How many meters is it to Andromeda, the nearest large galaxy, given that it is $2\text{.}\text{00}\times {\text{10}}^6$ light years away? (c) The most distant galaxy yet discovered is $\text{12}\text{.}0\times {\text{10}}^9$ light years away. How far is this in meters?

During normal beating, the heart creates a maximum 4.00-mV potential across 0.300 m of a person’s chest, creating a 1.00-Hz electromagnetic wave. (a) What is the maximum electric field strength created? (b) What is the corresponding maximum magnetic field strength in the electromagnetic wave? (c) What is the wavelength of the electromagnetic wave?

(a) What is the wavelength of 100-MHz radio waves used in an MRI unit? (b) If the frequencies are swept over a $\pm 1\text{.}\text{00}$ range centered on 100 MHz, what is the range of wavelengths broadcast?

TV-reception antennas for VHF are constructed with cross wires supported at their centers, as shown in Figure 24.26. The ideal length for the cross wires is one-half the wavelength to be received, with the more expensive antennas having one for each channel. Suppose you measure the lengths of the wires for particular channels and find them to be 1.94 and 0.753 m long, respectively. What are the frequencies for these channels?

Figure 24.26 A television reception antenna has cross wires of various lengths to most efficiently receive different wavelengths.

Lunar astronauts placed a reflector on the Moon’s surface, off which a laser beam is periodically reflected. The distance to the Moon is calculated from the round-trip time. (a) To what accuracy in meters can the distance to the Moon be determined, if this time can be measured to 0.100 ns? (b) What percent accuracy is this, given the average distance to the Moon is $3\text{.}\text{84}\times {\text{10}}^8\text{m}$?

Integrated Concepts

(a) Calculate the ratio of the highest to lowest frequencies of electromagnetic waves the eye can see, given the wavelength range of visible light is from 380 to 760 nm. (b) Compare this with the ratio of highest to lowest frequencies the ear can hear.

Find the intensity of an electromagnetic wave having a peak magnetic field strength of $4\text{.}\text{00}\times {\text{10}}^{-9}\text{T}$.

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 2.50-m-diameter university communications satellite dish receives TV signals that have a maximum electric field strength (for one channel) of $7\text{.}\text{50}\mu \mathrm{V/m}$. (See Figure 24.27.) (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 $1\text{.}\text{50}\times {\text{10}}^{\text{13}}{\text{m}}^2$ (a large fraction of North America), how much power does it radiate?

Figure 24.27 Satellite dishes receive TV signals sent from orbit. Although the signals are quite weak, the receiver can detect them by being tuned to resonate at their frequency.

Show that for a continuous sinusoidal electromagnetic wave, the peak intensity is twice the average intensity ($I_0={2I}_{\text{ave}}$), using either the fact that $E_0=\sqrt{2}{E}_{\text{rms}}$, or $B_0=\sqrt{2}{B}_{\text{rms}}$, where rms means average (actually root mean square, a type of average).

Integrated Concepts

An $\text{LC}$ circuit with a 5.00-pF capacitor oscillates in such a manner as to radiate at a wavelength of 3.30 m. (a) What is the resonant frequency? (b) What inductance is in series with the capacitor?

Integrated Concepts

Police radar determines the speed of motor vehicles using the same Doppler-shift technique employed for ultrasound in medical diagnostics. Beats are produced by mixing the double Doppler-shifted echo with the original frequency. If $1\text{.}\text{50}\times {\text{10}}^9\text{-Hz}$ microwaves are used and a beat frequency of 150 Hz is produced, what is the speed of the vehicle? (Assume the same Doppler-shift formulas are valid with the speed of sound replaced by the speed of light.)

Integrated Concepts

On its highest power setting, a microwave oven increases the temperature of 0.400 kg of spaghetti by $\text{45}\text{.}\mathrm{0º}\text{C}$ in 120 s. (a) What was the rate of power absorption by the spaghetti, given that its specific heat is $3\text{.}\text{76}\times {\text{10}}^3\text{J/kg}\cdot \text{ºC}$? (b) Find the average intensity of the microwaves, given that they are absorbed over a circular area 20.0 cm in diameter. (c) What is the peak electric field strength of the microwave? (d) What is its peak magnetic field strength?

Integrated Concepts

A 200-turn flat coil of wire 30.0 cm in diameter acts as an antenna for FM radio at a frequency of 100 MHz. The magnetic field of the incoming electromagnetic wave is perpendicular to the coil and has a maximum strength of $1\text{.}\text{00}\times {\text{10}}^{-\text{12}}\text{T}$. (a) What power is incident on the coil? (b) What average emf is induced in the coil over one-fourth of a cycle? (c) If the radio receiver has an inductance of $2\text{.}\text{50}\mu \mathrm{H}$, what capacitance must it have to resonate at 100 MHz?

Unreasonable Results

A researcher measures the wavelength of a 1.20-GHz electromagnetic wave to be 0.500 m. (a) Calculate the speed at which this wave propagates. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Unreasonable Results

An $\text{LC}$ circuit containing a 2.00-H inductor oscillates at such a frequency that it radiates at a 1.00-m wavelength. (a) What is the capacitance of the circuit? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Create Your Own ProblemConsider electromagnetic fields produced by high voltage power lines. Construct a problem in which you calculate the intensity of this electromagnetic radiation in ${\text{W/m}}^2$ based on the measured magnetic field strength of the radiation in a home near the power lines. Assume these magnetic field strengths are known to average less than a $\mu \mathrm{T}$. The intensity is small enough that it is difficult to imagine mechanisms for biological damage due to it. Discuss how much energy may be radiating from a section of power line several hundred meters long and compare this to the power likely to be carried by the lines. An idea of how much power this is can be obtained by calculating the approximate current responsible for $\mu \mathrm{T}$ fields at distances of tens of meters.

Create Your Own ProblemConsider the most recent generation of residential satellite dishes that are a little less than half a meter in diameter. Construct a problem in which you calculate the power received by the dish and the maximum electric field strength of the microwave signals for a single channel received by the dish. Among the things to be considered are the power broadcast by the satellite and the area over which the power is spread, as well as the area of the receiving dish.