what is the distance in m from the radar station to the reflecting object?

24.one Maxwell's Equations: Electromagnetic Waves Predicted and Observed

i .

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

two .

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.

24.2 Production of Electromagnetic Waves

iii.

What is the maximum electric field strength in an electromagnetic moving ridge that has a maximum magnetic field strength of $5\text{.}\text{00}\times {\text{10}}^{-4}\text{T}$ (nigh 10 times the Earth's)?

iv .

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

v .

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

24.three The Electromagnetic Spectrum

6.

(a) Two microwave frequencies are authorized for use in microwave ovens: 900 and 2560 MHz. Calculate the wavelength of each. (b) Which frequency would produce smaller hot spots in foods due to interference effects?

7 .

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

8.

A radio station utilizes frequencies between commercial AM and FM. What is the frequency of a eleven.12-m-wavelength channel?

9 .

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

x.

Combing your hair leads to backlog electrons on the comb. How fast would you have to move the comb upward and down to produce cerise light?

11 .

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

12.

Approximately what is the smallest detail appreciable with a microscope that uses ultraviolet light of frequency $1\text{.}\text{20}\times {\text{ten}}^{\text{15}}\text{Hz}$ ?

xiii .

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

14.

Some radar systems detect the size and shape of objects such as aircraft and geological terrain. Approximately what is the smallest observable detail utilizing 500-MHz radar?

xv .

Decide the corporeality of time it takes for X-rays of frequency $\mathrm{three}\times {\text{x}}^{\text{18}}\text{Hz}$ to travel (a) 1 mm and (b) 1 cm.

16.

If yous wish to observe details of the size of atoms (about $1\times {\text{10}}^{-\text{10}}\text{m}$ ) with electromagnetic radiation, information technology must have a wavelength of most this size. (a) What is its frequency? (b) What type of electromagnetic radiation might this exist?

17 .

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

18 .

Distances in infinite are often quoted in units of low-cal years, the distance lite travels in i year. (a) How many meters is a lite yr? (b) How many meters is it to Andromeda, the nearest large milky way, given that it is $2\text{.}\text{00}\times {\text{10}}^6$ light years away? (c) The most distant galaxy all the same discovered is $\text{12}\text{.}0\times {\text{10}}^9$ light years away. How far is this in meters?

19.

A certain 50.0-Hz AC electric line radiates an electromagnetic wave having a maximum electric field strength of xiii.0 kV/m. (a) What is the wavelength of this very depression frequency electromagnetic wave? (b) What is its maximum magnetic field strength?

20 .

During normal chirapsia, the heart creates a maximum iv.00-mV potential across 0.300 m of a person'due south chest, creating a 1.00-Hz electromagnetic moving ridge. (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?

21.

(a) The ideal size (near efficient) for a broadcast antenna with 1 end on the ground is one-quaternary the wavelength ( $\lambda /4$ ) of the electromagnetic radiation being sent out. If a new radio station has such an antenna that is 50.0 yard high, what frequency does it circulate almost efficiently? Is this in the AM or FM band? (b) Discuss the analogy of the fundamental resonant mode of an air column closed at one terminate to the resonance of currents on an antenna that is one-4th their wavelength.

22 .

(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?

23.

(a) What is the frequency of the 193-nm ultraviolet radiation used in laser heart surgery? (b) Assuming the accuracy with which this EM radiation can ablate the cornea is directly proportional to wavelength, how much more accurate can this UV be than the shortest visible wavelength of lite?

24 .

Idiot box-reception antennas for VHF are synthetic with cross wires supported at their centers, as shown in Figure 24.28. The ideal length for the cross wires is half the wavelength to be received, with the more expensive antennas having one for each channel. Suppose you mensurate the lengths of the wires for detail channels and find them to be one.94 and 0.753 m long, respectively. What are the frequencies for these channels?

Figure 24.28 A tv set reception antenna has cross wires of various lengths to well-nigh efficiently receive different wavelengths.

25.

Conversations with astronauts on lunar walks had an echo that was used to estimate the distance to the Moon. The sound spoken past the person on Earth was transformed into a radio signal sent to the Moon, and transformed back into sound on a speaker inside the astronaut's space suit. This sound was picked up by the microphone in the space arrange (intended for the astronaut's voice) and sent back to Earth as a radio repeat of sorts. If the circular-trip time was two.60 s, what was the approximate distance to the Moon, neglecting whatsoever delays in the electronics?

26 .

Lunar astronauts placed a reflector on the Moon'southward surface, off which a light amplification by stimulated emission of radiation axle is periodically reflected. The distance to the Moon is calculated from the round-trip fourth dimension. (a) To what accuracy in meters tin can the distance to the Moon be determined, if this time tin exist measured to 0.100 ns? (b) What percent accuracy is this, given the average distance to the Moon is $3\text{.}\text{84}\times {\text{10}}^{\mathrm{eight}}\text{m}$ ?

27.

Radar is used to determine distances to diverse objects by measuring the round-trip time for an repeat from the object. (a) How far away is the planet Venus if the echo time is one thousand s? (b) What is the repeat time for a machine 75.0 m from a Highway Police radar unit of measurement? (c) How accurately (in nanoseconds) must yous be able to measure the echo time to an plane 12.0 km abroad to determine its distance within 10.0 g?

28 .

Integrated Concepts

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

29.

Integrated Concepts

(a) Calculate the charge per unit in watts at which rut transfer through radiation occurs (nearly entirely in the infrared) from $\mathrm{i}\text{.}0{\text{m}}^2$ of the Earth's surface at night. Assume the emissivity is 0.90, the temperature of the World is $\text{15º}\text{C}$ , and that of outer space is 2.seven K. (b) Compare the intensity of this radiation with that coming to the Earth from the Sun during the day, which averages about $\text{800}{\text{West/m}}^2$ , simply half of which is absorbed. (c) What is the maximum magnetic field forcefulness in the outgoing radiation, assuming information technology is a continuous wave?

24.iv Energy in Electromagnetic Waves

30.

What is the intensity of an electromagnetic wave with a peak electrical field forcefulness of 125 Five/m?

31 .

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

32.

Assume the helium-neon lasers unremarkably used in student physics laboratories have power outputs of 0.250 mW. (a) If such a light amplification by stimulated emission of radiation axle is projected onto a round spot ane.00 mm in diameter, what is its intensity? (b) Discover the peak magnetic field forcefulness. (c) Find the peak electric field forcefulness.

33 .

An AM radio transmitter broadcasts 50.0 kW of power uniformly in all directions. (a) Bold all of the radio waves that strike the basis are completely absorbed, and that in that location is no absorption by the temper or other objects, what is the intensity xxx.0 km abroad? (Hint: One-half the power volition exist spread over the area of a hemisphere.) (b) What is the maximum electrical field force at this distance?

34.

Suppose the maximum prophylactic intensity of microwaves for human exposure is taken to exist $1\text{.}\text{00 W}{\text{/m}}^2$ . (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 lot be to exist exposed to an intensity considered to be safe? Assume that the ability spreads uniformly over the area of a sphere with no complications from absorption or reflection. (b) What is the maximum electric field forcefulness at the rubber intensity? (Note that early radar units leaked more modern ones exercise. This caused identifiable health problems, such every bit cataracts, for people who worked virtually them.)

35 .

A 2.l-m-bore academy communications satellite dish receives Television receiver signals that take a maximum electric field strength (for 1 channel) of $7\text{.}\text{50}\mu \mathrm{V/g}$ . (Encounter Effigy 24.29.) (a) What is the intensity of this wave? (b) What is the power received past the antenna? (c) If the orbiting satellite broadcasts uniformly over an area of $\mathrm{one}\text{.}\text{50}\times {\text{10}}^{\text{13}}{\text{yard}}^{\mathrm{ii}}$ (a large fraction of Northward America), how much power does it radiate?

Effigy 24.29 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.

36.

Lasers tin can be constructed that produce an extremely loftier intensity electromagnetic wave for a brief time—called pulsed lasers. They are used to ignite nuclear fusion, for case. Such a laser may produce an electromagnetic wave with a maximum electrical field strength of $1\text{.}\text{00}\times {\text{10}}^{\text{11}}\text{V}/\text{1000}$ for a time of 1.00 ns. (a) What is the maximum magnetic field force in the wave? (b) What is the intensity of the beam? (c) What energy does information technology evangelize on a $\mathrm{i}\text{.}\text{00}{\text{-mm}}^{\mathrm{two}}$ surface area?

37 .

Show that for a continuous sinusoidal electromagnetic moving ridge, the height intensity is twice the average intensity ( $I_0={\mathrm{ii}I}_{\text{ave}}$ ), using either the fact that ${\mathrm{East}}_0=\sqrt{2}{E}_{\text{rms}}$ , or $B_0=\sqrt{2}{B}_{\text{rms}}$ , where rms means average (actually root hateful square, a blazon of boilerplate).

38.

Suppose a source of electromagnetic waves radiates uniformly in all directions in empty infinite where in that location are no absorption or interference effects. (a) Testify that the intensity is inversely proportional to $r^2$ , the distance from the source squared. (b) Show that the magnitudes of the electric and magnetic fields are inversely proportional to $r$ .

39 .

Integrated Concepts

An $\text{LC}$ circuit with a 5.00-pF capacitor oscillates in such a style equally to radiate at a wavelength of 3.thirty yard. (a) What is the resonant frequency? (b) What inductance is in series with the capacitor?

xl.

Integrated Concepts

What capacitance is needed in series with an $\text{800}-\mu \text{H}$ inductor to grade a circuit that radiates a wavelength of 196 thousand?

41 .

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 $1\text{.}\text{50}\times {\text{10}}^{\mathrm{ix}}\text{-Hz}$ 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 audio replaced by the speed of lite.)

42.

Integrated Concepts

Assume the mostly infrared radiations from a heat lamp acts like a continuous moving ridge with wavelength $\mathrm{i.50}\mu \text{one thousand}$ . (a) If the lamp's 200-Westward output is focused on a person's shoulder, over a round surface area 25.0 cm in diameter, what is the intensity in ${\text{Westward/m}}^2$ ? (b) What is the summit electric field strength? (c) Find the elevation magnetic field strength. (d) How long will information technology take to increase the temperature of the iv.00-kg shoulder by $\mathrm{2.00º\; C}$ , assuming no other heat transfer and given that its specific heat is $3\text{.}\text{47}\times {\text{x}}^3\text{J/kg}\cdot \text{ºC}$ ?

43 .

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 south. (a) What was the rate of power absorption by the spaghetti, given that its specific heat is $3\text{.}\text{76}\times {\text{ten}}^{\mathrm{iii}}\text{J/kg}\cdot \text{ºC}$ ? (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 force?

44.

Integrated Concepts

Electromagnetic radiation from a 5.00-mW light amplification by stimulated emission of radiation is full-bodied on a $1\text{.}\text{00}{\text{-mm}}^2$ area. (a) What is the intensity in ${\text{W/m}}^2$ ? (b) Suppose a 2.00-nC static charge is in the beam. What is the maximum electric force it experiences? (c) If the static charge moves at 400 thou/due south, what maximum magnetic strength tin information technology feel?

45 .

Integrated Concepts

A 200-plough flat roll of wire xxx.0 cm in bore 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 ringlet and has a maximum force of $1\text{.}\text{00}\times {\text{10}}^{-\text{12}}\text{T}$ . (a) What ability is incident on the roll? (b) What average emf is induced in the coil over i-fourth of a bike? (c) If the radio receiver has an inductance of $2\text{.}\text{50}\mu \mathrm{H}$ , what capacitance must it accept to resonate at 100 MHz?

46.

Integrated Concepts

If electric and magnetic field strengths vary sinusoidally in fourth dimension, beingness zero at $t=0$ , then $\mathrm{Due\; east}=E_0\text{sin}\mathrm{2\pi }\text{ft}$ and $B=B_0\text{sin}\mathrm{2\pi }\text{ft}$ . Let $f=1\text{.}\text{00 GHz}$ here. (a) When are the field strengths first zero? (b) When practise they reach their near negative value? (c) How much time is needed for them to complete one cycle?

47 .

Unreasonable Results

A researcher measures the wavelength of a 1.xx-GHz electromagnetic moving ridge to be 0.500 m. (a) Summate the speed at which this wave propagates. (b) What is unreasonable almost this upshot? (c) Which assumptions are unreasonable or inconsistent?

48.

Unreasonable Results

The tiptop magnetic field strength in a residential microwave oven is $\mathrm{nine}\text{.}\text{xx}\times {\text{ten}}^{-\mathrm{five}}\text{T}$ . (a) What is the intensity of the microwave? (b) What is unreasonable about this outcome? (c) What is wrong well-nigh the premise?

49 .

Unreasonable Results

An $\text{LC}$ circuit containing a ii.00-H inductor oscillates at such a frequency that it radiates at a i.00-m wavelength. (a) What is the capacitance of the excursion? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

50.

Unreasonable Results

An $\text{LC}$ circuit containing a 1.00-pF capacitor oscillates at such a frequency that it radiates at a 300-nm wavelength. (a) What is the inductance of the circuit? (b) What is unreasonable well-nigh this result? (c) Which assumptions are unreasonable or inconsistent?

51 .

Create Your Own Problem

Consider electromagnetic fields produced by high voltage power lines. Construct a trouble in which you calculate the intensity of this electromagnetic radiation in ${\text{Westward/m}}^{\mathrm{ii}}$ based on the measured magnetic field strength of the radiation in a dwelling house nearly 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 hard to imagine mechanisms for biological damage due to it. Hash out how much free energy may be radiating from a department of electric line several hundred meters long and compare this to the power probable to be carried by the lines. An idea of how much power this is can be obtained past calculating the judge current responsible for $\mu \mathrm{T}$ fields at distances of tens of meters.

52 .

Create Your Own Problem

Consider the nigh recent generation of residential satellite dishes that are a picayune less than one-half a meter in bore. Construct a trouble in which you calculate the power received by the dish and the maximum electrical field strength of the microwave signals for a single channel received by the dish. Among the things to be considered are the power broadcast past the satellite and the expanse over which the ability is spread, as well as the area of the receiving dish.

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Source: https://openstax.org/books/college-physics-ap-courses/pages/24-problems-exercises