Problem 33
Suppose an RLC circuit in resonance is used to produce a radio wave of wavelength \(150 \mathrm{~m}\). If the circuit has a 2.0 -pF capacitor, what size inductor is used?
Problem 34
Three FM radio stations covering the same geographical area broadcast at frequencies \(91.1,91.3,\) and \(91.5 \mathrm{MHz},\) respectively. What is the maximum allowable wavelength width of the band-pass filter in a radio receiver so that the FM station 91.3 can be played free of interference from FM 91.1 or FM 91.5? Use \(c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\), and calculate the wavelength to an uncertainty of \(1 \mathrm{~mm} .\)
Problem 35
A monochromatic point source of light emits \(1.5 \mathrm{~W}\) of electromagnetic power uniformly in all directions. Find the Poynting vector at a point situated at each of the following locations: a) \(0.30 \mathrm{~m}\) from the source b) \(0.32 \mathrm{~m}\) from the source c) \(1.00 \mathrm{~m}\) from the source
Problem 36
Consider an electron in a hydrogen atom, which is \(0.050 \mathrm{nm}\) from the proton in the nucleus. a) What electric field does the electron experience? b) In order to produce an electric field whose root-meansquare magnitude is the same as that of the field in part (a), what intensity must a laser light have?
Problem 39
Calculate the average value of the Poynting vector, \(S_{\text {ave }}\) for an electromagnetic wave having an electric field of amplitude \(100 . \mathrm{V} / \mathrm{m}\) a) What is the average energy density of this wave? b) How large is the amplitude of the magnetic field?
Problem 40
The most intense beam of light that can propagate through dry air must have an electric field whose maximum amplitude is no greater than the breakdown value for air: \(E_{\max }^{\operatorname{air}}=3.0 \cdot 10^{6} \mathrm{~V} / \mathrm{m},\) assuming that this value is unaffected by the frequency of the wave. a) Calculate the maximum amplitude the magnetic field of this wave can have. b) Calculate the intensity of this wave. c) What happens to a wave more intense than this?
Problem 41
A continuous-wave (cw) argon-ion laser beam has an average power of \(10.0 \mathrm{~W}\) and a beam diameter of \(1.00 \mathrm{~mm}\). Assume that the intensity of the beam is the same throughout the cross section of the beam (which is not true, as the actual distribution of intensity is a Gaussian function). a) Calculate the intensity of the laser beam. Compare this with the average intensity of sunlight at Earth's surface \(\left(1400 . \mathrm{W} / \mathrm{m}^{2}\right)\) b) Find the root-mean-square electric field in the laser beam. c) Find the average value of the Poynting vector over time. d) If the wavelength of the laser beam is \(514.5 \mathrm{nm}\) in vacuum, write an expression for the instantaneous Poynting vector, where the instantaneous Poynting vector is zero at \(t=0\) and \(x=0\) e) Calculate the root-mean-square value of the magnetic field in the laser beam.
Problem 42
A voltage, \(V\), is applied across a cylindrical conductor of radius \(r\), length \(L\), and resistance \(R\). As a result, a current, \(i\), is flowing through the conductor, which gives rise to a magnetic field, \(B\). The conductor is placed along the \(y\) -axis, and the current is flowing in the positive \(y\) -direction. Assume that the electric field is uniform throughout the conductor. a) Find the magnitude and the direction of the Poynting vector at the surface of the conductor. b) Show that \(\int \vec{S} \cdot d \vec{A}=i^{2} R\)
Problem 44
Scientists have proposed using the radiation pressure of sunlight for travel to other planets in the Solar System. If the intensity of the electromagnetic radiation produced by the Sun is about \(1.40 \mathrm{~kW} / \mathrm{m}^{2}\) near the Earth, what size would a sail have to be to accelerate a spaceship with a mass of 10.0 metric tons at \(1.00 \mathrm{~m} / \mathrm{s}^{2} ?\) a) Assume that the sail absorbs all the incident radiation. b) Assume that the sail perfectly reflects all the incident radiation.
Problem 45
A solar sail is a giant circle (with a radius \(R=10.0 \mathrm{~km}\) ) made of a material that is perfectly reflecting on one side and totally absorbing on the other side. In deep space, away from other sources of light, the cosmic microwave background will provide the primary source of radiation incident on the sail. Assuming that this radiation is that of an ideal black body at \(T=2.725 \mathrm{~K},\) calculate the net force on the sail due to its reflection and absorption.
