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optics

Optics 5th edition Eugene Hecht - Solutions

Figure P.4.3 depicts light emerging from a point source. It shows three different representations of radiant energy streaming outward. Identify each one and discuss its relationship to the others.Figure P.4.3 シ
A white floodlight beam crosses a large volume containing a tenuous molecular gas mixture of mostly oxygen and nitrogen. Compare the relative amount of scattering occurring for the yellow (580 nm) component with that of the violet (400 nm) component.
Work your way through an argument using dimensional analysis to establish the λ-4 dependence of the percentage of light scattered in Rayleigh Scattering. Let E0i and E0s be the incident and scattered amplitudes, the latter at a distance r from the scatterer. Assume E0s ∝ E0i and E0s ∝ 1/r.
If an ultraviolet photon is to dissociate the oxygen and carbon atoms in the carbon monoxide molecule, it must provide 11 eV of energy. What is the minimum frequency of the appropriate radiation?
In 1871 Sellmeier derived the equationwhere the Aj terms are constants and each Î»0j is the vacuum wavelength associated with a natural frequency v0j, such that Î»0jv0j = c. This formulation is a considerable practical improvement over the Cauchy Equation. Show
Crystal quartz has refractive indexes of 1.557 and 1.547 at wavelengths of 410.0 nm and 550.0 nm, respectively. Using only the first two terms in Cauchy’s Equation, calculate C1 and C2 and determine the index of refraction of quartz at 610.0 nm.
Referring to the previous problem, realize that there is a region between each pair of absorption bands for which the Cauchy Equation (with a new set of constants) works fairly well. Examine Fig. 3.41: what can you say about the various values of C1as Ï‰ decreases across the spectrum?
Augustin Louis Cauchy (1789€“1857) determined an empirical equation for n(Î») for substances that are transparent in the visible. His expression corresponded to the power series relationwhere the Cs are all constants. In light of Fig. 3.41, what is the physical significance of
Show that Eq. (3.70) can be rewritten aswhere C = 4Ï€2c2Ïµ0 me/Nq2e. |(1² – 1)1 = -Cx? + CAq²
The resonant frequency of lead glass is in the UV fairly near the visible, whereas that for fused silica is far into the UV. Use the dispersion equation to make a rough sketch of n versus ω for the visible region of the spectrum.
Take Eq. (3.71) and check out the units to make sure that they agree on both sides. Nq? n°(m) = 1 + €o me (3.71) lWój – w² @õj
Fuchsin is a strong (aniline) dye, which in solution with alcohol has a deep red color. It appears red because it absorbs the green component of the spectrum. (As you might expect, the surfaces of rystals of fuchsin reflect green light rather strongly.) Imagine that you have a thin-walled
In the next chapter, Eq. (4.47), we€™ll see that a substance reflects radiant energy appreciably when its index differs most from the medium in which it is embedded.(a) The dielectric constant of ice measured at microwave frequencies is roughly 1, whereas that for water is about 80 times
Show that for substances of low density, such as gases, which have a single resonant frequency Ï‰0, the index of refraction is given by Ng. 2eom,(w3 – w) n = 1 +
The low-frequency relative permittivity of water varies from 88.00 at 0°C to 55.33 at 100°C. Explain this behavior. Over the same range in temperature, the index of refraction (λ = 589.3 nm) goes from roughly 1.33 to 1.32. Why is the change in n so much smaller than the corresponding change in
A lightwave travels from point A to point B in vacuum. Suppose we introduce into its path a flat glass plate (ng = 1.50) of thickness L = 1.00 mm. If the vacuum wavelength is 500 nm, how many waves span the space from A to B with and without the glass in place? What phase shift is introduced with
Yellow light from a sodium lamp (λ0 = 589 nm) traverses a tank of glycerin (of index 1.47), which is 20.0 m long, in a time t1. If it takes a time t2 for the light to pass through the same tank when filled with carbon disulfide (of index 1.63), determine the value of t2 - t1.
A 500-nm lightwave in vacuum enters a glass plate of index 1.60 and propagates perpendicularly across it. How many waves span the glass if it’s 1.00 cm thick?
What is the distance that yellow light travels in water (where n = 1.33) in 1.00 s?
If the speed of light (the phase speed) in Fabulite (SrTiO3) is 1.245 × 108 m/s, what is its index of refraction?
Determine the index of refraction of a medium if it is to reduce the speed of light by 10% as compared to its speed in vacuum.
Given that the wavelength of a lightwave in vacuum is 540 nm, what will it be in water, where n = 1.33?
What is the speed of light in diamond if the index of refraction is 2.42?
The time average of some function Æ’(t) taken over an interval T is given bywhere t' is just a dummy variable. If Ï„ = 2Ï€/Ï‰ is the period of a harmonic function, show that vector r.andwhen T = Ï„ and when T > > Ï„. ri+T f(t') dt'
Using the wave given in the previous problem, determine vector E(-λ/2, 0) and draw a sketch of the vector representing it at that moment.
A plane, harmonic, linearly polarized lightwave has an electric field intensity given bywhile traveling in a piece of glass. Find(a) The frequency of the light.(b) Its wavelength.(c) The index of refraction of the glass. E; = Eo cosT 1015 0.65c
Consider the uniformly moving charge depicted in Fig. 3.26b. Draw a sphere surrounding it and show via the Poynting vector that the charge does not radiate.Fig. 3.26b E
Consider the plight of an astronaut floating in free space with only a 10-W lantern (inexhaustibly supplied with power). How long will it take to reach a speed of 10 m/s using the radiation as propulsion? The astronaut’s total mass is 100 kg.
A parabolic radar antenna with a 2-m diameter transmits 200-kW pulses of energy. If its repetition rate is 500 pulses per second, each lasting 2 μs, determine the average reaction force on the antenna.
What force on the average will be exerted on the (40 m × 50 m) flat, highly reflecting side of a space station wall if it’s facing the Sun while orbiting Earth?
Consider an electromagnetic wave impinging on an electron. It is easy to show kinematically that the average value of the time rate-of-change of the electron€™s momentum vector p is proportional to the average value of the time rate-of-change of the work, W, done on it by the wave. In
A tungsten lightbulb puts out 20 W of radiant energy (most of it IR). Assume it to be a point source and calculate the irradiance 1.00m away.
The following is the expression for the vector E-field of an electromagnetic wave traveling in a homogeneous dielectric:Here v = 1.80 Ã— 1015 rad/s and k = 1.20 Ã— 107 rad/m.(a) Determine the associated vector B-field.(b) Find the index of refraction.(c) Compute the
A light beam with an irradiance of 2.00 × 106 W/m2 impinges normally on a surface that reflects 70.0% and absorbs 30.0%. Compute the resulting radiation pressure on the surface.
A surface is placed perpendicular to a beam of light of constant irradiance (I). Suppose that the fraction of the irradiance absorbed by the surface is α. Show that the pressure on the surface is given byP = (2 - α)I/c
The average magnitude of the Poynting vector for sunlight arriving at the top of Earth’s atmosphere (1.5 × 1011 m from the Sun) is about 1.4 kW/m2.(a) Compute the average radiation pressure exerted on a metal reflector facing the Sun.(b) Approximate the average radiation pressure at the surface
A completely absorbing screen receives 300 W of light for 100 s. Compute the total linear momentum transferred to the screen.
Derive an expression for the radiation pressure when the normally incident beam of light is totally reflected. Generalize this result to the case of oblique incidence at an angle θ with the normal.
What is the momentum of a 1019-Hz X- ray photon?
A cloud of locusts having a density of 100 insects per cubic meter is flying north at a rate of 6 m/min. What is the flux density of locusts? That is, how many cross an area of 1 m2 perpendicular to their flight path per second?
A nearly cylindrical laserbeam impinges normally on a perfectly absorbing surface. The irradiance of the beam (assuming it to be uniform over its cross section) is 40 W/cm2. If the diameter of the beam is 2.0/√π cm how much energy is absorbed per minute?
Prove that the irradiance of a harmonic EM wave in vacuum is given byand then determine the average rate at which energy is transported per unit area by a plane wave having an amplitude of 15.0 V/m. -Еб 2спо
Starting with Eq. (3.32), prove that the energy densities of the electric and magnet fields are equal (uE= uB) for an electromagnetic wave. (3.32) UB 2µo
The E-field of an electromagnetic wave is described byWrite an expression for the B-field. Determine vector B(0, 0). Ё 3 (i + jEo sin (kz — ot + п/6)
If the electric field vector E(z, t) of an EM wave in vacuum is, at a certain location and time, given by vector E = (10 V/m)(cos 0.5π) iˆ write an expression for the associated vector B-field.
An electromagnetic wave is specified (in SI units) by the following function:Remember that vector E0 and vector k are perpendicular to each other. Find (a) The direction along which the electric field oscillates(b) The scalar value of amplitude of the electric field(c) The direction of
A linearly polarized harmonic plane wave with a scalar amplitude of 10 V/m is propagating along line in the xy-plane at 45° to the x-axis with the xy-plane as its plane of vibration. Please write a vector expression describing the wave assuming both kx and ky are positive. Calculate the flux
A 1.0-mW laser has a beam diameter of 2 mm. Assuming the divergence of the beam to be negligible, compute its energy density in the vicinity of the laser.
How many photons per second are emitted from a 100-W yellow lightbulb if we assume negligible thermal losses and a quasimonochromatic wavelength of 550 nm? In actuality only about 2.5% of the total dissipated power emerges as visible radiation in an ordinary 100-W incandescent lamp.
A harmonic electromagnetic plane wave with a wavelength of 0.12 m travels in vacuum in the positive z-direction. It oscillates along the x-axis such that at t = 0 and z = 0, the E-field has a maximum value of E(0, 0) = +6.0 V/m.(a) Write an expression for vector E(z, t).(b) Write an expression for
An isotropic quasimonochromatic point source radiates at a rate of 100 W. What is the flux density at a distance of 1 m? What are the amplitudes of the vector E- and vector B-fields at that point?
Using energy arguments, show that the amplitude of a cylindrical wave must vary inversely with √r. Draw a diagram indicating what’s happening.
A 3.0-V incandescent flashlight bulb draws 0.25 A, converting about 1.0% of the dissipated power into light (λ ≈ 550 nm). If the beam has a cross-sectional area of 10 cm2 and is approximately cylindrical(a) How many photons are emitted per second?(b) How many photons occupy each meter of the
Imagine that you are standing in the path of an antenna that is radiating plane waves of frequency 100 MHz and flux density 19.88 × 10-2 W/m-2. Compute the photon flux density, that is, the number of photons per unit time per unit area. How many photons, on the average, will be found in a
A laser provides pulses of EM-radiation in vacuum lasting 10-12 s. If the radiant flux density is 1020 W/m2, determine the amplitude of the electric field of the beam.
Pulses of UV lasting 2.00 ns each are emitted from a laser that has a beam of diameter 2.5 mm. Given that each burst carries an energy of 6.0 J,(a) determine the length in space of each wavetrain.(b) find the average energy per unit volume for such a pulse.
On average, the net electromagnetic power radiated by the Sun, its so-called luminosity (L), is 3.9 × 1026 W. Determine the mean amplitude of the electric field due to all the radiant energy arriving at the top of Earth’s atmosphere (1.5 × 1011 m from the Sun).
Consider a linearly polarized plane electromagnetic wave traveling in the +x-direction in free space having as its plane of vibration the xy-plane. Given that its frequency is 10 MHz and its amplitude is E0 = 0.08 V/m,(a) Find the period and wavelength of the wave.(b) Write an expression for E(t)
A 1.0-mW laser produces a nearly parallel beam 1.0 cm2 in cross-sectional area at a wavelength of 650 nm. Determine the amplitude of the electric field in the beam, assuming the wavefronts are homogeneous and the light travels in vacuum.
With the previous problem in mind, prove thatfor any interval T. |(sin*wt)T = 1– sinc wT cos 2wt]
Show that a more general formulation of the previous problem yieldsfor any interval T. (cos*ot)T = [1 + sinc @T cos 2ot]
Calculate the energy input necessary to charge a parallel-plate capacitor by carrying charge from one plate to the other. Assume the energy is stored in the field between the plates and compute the energy per unit volume, uE, of that region, that is, Eq. (3.31). Hint: Since the electric field
Given that the vector B-field of an electromagnetic wave in vacuum iswrite an expression for the associated vector E-field. What is the direction of propagation? i(kz+@t) Вс, у. г, 1) %— Вoјенletan)
A plane electromagnetic wave traveling in the y-direction through vacuum is given byDetermine an expression for the corresponding magnetic field of the electromagnetic wave. Draw a diagram showing vector E0, vector B0, and vector k, the propagation vector. É(x, y, z, t) = Eſîeky-ws)
A 550-nm harmonic EM wave whose electric field is in the z-direction is traveling in the y-direction in vacuum.(a) What is the frequency of the wave?(b) Determine both ω and k for this wave.(c) If the electric field amplitude is 600 V/m, what is the amplitude of the magnetic field?(d) Write an
The electric field of an electromagnetic wave traveling in the positive x-direction is given by(a) Describe the field verbally.(b) Determine an expression for k.(c) Find the phase speed of the wave. Ё 3 Еoj sin TTZ -cos (kx – wt)
Imagine an electromagnetic wave with its vector E-field in the y-direction. Show that Eq. (3.27) ДЕ дв дх дt ||
Considering Eq. (3.30), show that the expressionis correct as it applies to a plane wave for which the direction of the electric field is constant. KxE= ωΒ (3.30) Ey 3 св.
Write an expression for the vector E- and vector B-fields that constitute a plane harmonic wave traveling in the +z-direction. The wave is linearly polarized with its plane of vibration at 45° to the yz-plane.
Consider the plane electromagnetic wave in vacuum (in SI units) given by the expressions Ex = 0, Ey = 2 cos [2π × 1014(t - x/c) + π/2], and Ez = 0.(a) What are the frequency, wavelength, direction of motion, amplitude, initial phase angle, and polarization of the wave?(b) Write an expression for
The profile of a transverse harmonic wave, traveling at 1.2 m/s on a string, is given byy = (0.02 m) sin (157 m-1) xDetermine its amplitude, wavelength, frequency, and period.
Which of the following expressions correspond to traveling waves? For each of those, what is the speed of the wave? The quantities α, b, and c are positive constants.(a) ψ(z, t) = (αz - bt)2(b) ψ(x, t) = (αx + bt + c)2
Make up a table with columns headed by values of kx running from x = -λ/2 to x = +λ in intervals of x of λ/4. In each column place the corresponding values of cos kx and beneath that the values of cos (kx + π). Next plot the three functions cos kx, cos (kx + π), and cos kx + cos (kx + π).
Create an expression for the profile of a harmonic wave traveling in the z-direction whose magnitude at z = -λ/12 is 0.866, at z = +λ/6 is 1/2, and at z = λ/4 is 0.
Does the following function, in which A is a constant,Ψ(y, t) = (y - vt)Arepresent a wave? Explain your reasoning.
Show that the functionis a nontrivial solution of the differential wave equation. In what direction does it travel? (z + vt U(Z, t) (12 + 2) = (1 °2)p
Show that the functionis a solution of the differential wave equation. In what direction does it travel? (y, t) = (y – 41)²
Consider the functionwhere A is a constant. Show that it is a solution of the differential wave equation. Determine the speed of the wave and the direction of propagation. = (1 '2)p (z – vt)² + 1
Argon-ion lasers typically generate multi-watt beams in the green or blue regions of the visible spectrum. Determine the frequency of such a 514.5-nm beam.
Establish thatwhere A, Î±, b, and c are all constant, is a solution of the differential wave equation. This is a Gaussian or bell-shaped function. What is its speed and direction of travel? 4(y, t) = Ae¯a(by–ct)?
How many “yellow” lightwaves (λ = 580 nm) will fit into a distance in space equal to the thickness of a piece of paper (0.003 in.)? How far will the same number of microwaves (v = 1010 Hz, i.e., 10 GHz, and v = 3 × 108 m/s) extend?
The speed of light in vacuum is approximately 3 × 108 m/s. Find the wavelength of red light having a frequency of 5 × 1014 Hz. Compare this with the wavelength of a 60-Hz electromagnetic wave.
It is possible to generate ultrasonic waves in crystals with wavelengths similar to those of light (5 × 10-5 cm) but with lower frequencies (6 × 108 Hz). Compute the corresponding speed of such a wave.
A youngster in a boat on a lake watches waves that seem to be an endless succession of identical crests passing with a half-second interval between each. If every disturbance takes 1.5 s to sweep straight along the length of her 4.5-m-long boat, what are the frequency, period, and wavelength of the
A vibrating hammer strikes the end of a long metal rod in such a way that a periodic compression wave with a wavelength of 4.3 m travels down the rod’s length at a speed of 3.5 km/s. What was the frequency of the vibration?
A violin is submerged in a swimming pool at the wedding of two scuba divers. Given that the speed of compression waves in pure water is 1498 m/s, what is the wavelength of an A-note of 440 Hz played on the instrument?
A wavepulse travels 10 m along the length of a string in 2.0 s. A harmonic disturbance of wavelength 0.50 m is then generated on the string. What is its frequency?
Show that for a periodic wave ω = (2π/λ)v.
Make up a table with columns headed by values of θ running from -π/2 to 2π in intervals of π/4. In each column place the corresponding value of sin θ, beneath those the values of cos θ, beneath those the values of sin (θ - π/4), and similarly with the functions sin (θ - π/2), sin (θ -
Make up a table with columns headed by values of kx running from x = -λ/2 to x = +λ in intervals of x of λ/4 - of course, k = 2π/λ. In each column place the corresponding values of cos (kx - π/4) and beneath that the values of cos (kx + 3π/4). Next plot the functions 15 cos (kx - π/4) and
Make up a table with columns headed by values of ωt running from t = -τ/2 to t = +τ in intervals of t of τ/4— of course, ω = 2π/τ. In each column place the corresponding values of sin (ωτ + π/4) and sin (π/4 - ωt) and then plot these two functions.
Figure P.2.18 represents the profile (t = 0) of a transverse wave on a string traveling in the positive x-direction at a speed of 20.0 m/s.(a) Determine its wavelength.(b) What is the frequency of the wave?(c) Write down the wavefunction for the disturbance.Figure P.2.18 (х, Os) (m) 0.020 0.010
Figure P.2.19 represents the profile (t = 0) of a transverse wave on a string traveling in the positive z-direction at a speed of 100 cm/s.(a) Determine its wavelength.(b) What is the frequency of the wave? (z, Os) (cm) Figure P.2.19 20.0- 10.0 20.0 -20.0 -10.0 10.0 30.0 40.0 - 10.0 z (cm) -20.0
A transverse wave on a string travels in the negative y-direction at a speed of 40.0 cm/s. Figure P.2.20 is a graph of Ïˆ versus t showing how a point on the rope at y = 0 oscillates.(a) Determine the wave€™s period.(b) What is the frequency of the wave?(c) What is the
Given the wavefunctionsanddetermine in each case the values of(a) Frequency(b) Wavelength(c) period(d) amplitude(e) Phase velocity(f) Direction of motion.Time is in seconds and x is in meters. – 3t) 1 = 4 sin 2T (0.2x sin (7x + 3.5t) 2.5
The wavefunction of a transverse wave on a string isψ(x, t) = (30.0 cm) cos [(6.28 rad/m)x - (20.0 rad/s)t]Compute the (a) Frequency(b) Wavelength(c) Period(d) Amplitude,(e) Phase velocity(f) Direction of motion.
A traveling wave is given in SI units by the expressionFind its(a) Amplitude(b) Frequency(c) Wavelength(d) Speed(e) Period(f) Direction of propagation. Фу, t) —D 10sin 2т(5.0 X 1014 \3.0 × 108 $Фу, t) —D 10sin 2т(5.0 X 1014 \3.0 × 108$
Show that ψ(x, t) = A sin k(x - vt)is a solution of the differential wave equation.
Show thatψ(x, t) = A cos (kx - ωt)is a solution of the differential wave equation.
Prove thatψ(x, t) = A cos (kx - ωt - π/2)is equivalent toψ(x, t) = A sin (kx - ωt)

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