32. Electromagnetic Waves

A standard cell phone transmits electromagnetic waves with a frequency of $1.90\times10^9Hz$ . Calculate the wavelength of these electromagnetic waves (in cm).

(I) What is the range of wavelengths for (a) FM radio (88 MHz to 108 MHz) and (b) AM radio (535 kHz to 1700 kHz)?

(I) An EM wave has frequency 2.65 x 10¹⁴ Hz. What is its wavelength, and how would we classify it?

(III) Stars located in a certain cluster are assumed to be about the same distance from us. Two such stars have spectra that peak at λ₁ = 470nm and λ₂ = 720 nm, and the ratio of their apparent brightness is b₁/b₂ = 0.091. Estimate their relative sizes (give ratio of their diameters) using Wien’s law and the Stefan-Boltzmann equation, Eq. 19–17.

In free space (“vacuum”), where the net charge and current flow is zero, the speed of an EM wave is given by v = 1 √ε₀μ₀. If, instead, an EM wave travels in a nonconducting (“dielectric”) material with dielectric constant K, then v = 1 √Kε₀μ₀. For frequencies corresponding to the visible spectrum (near 5 x 10¹⁴ Hz), the dielectric constant of water is 1.77. Predict the speed of light in water and compare this value (as a percentage) with the speed of light in a vacuum.

(I) (a) What is the wavelength of a 35.75 x 10⁹-Hz radar signal? (b) What is the frequency of an X-ray with wavelength 0.12 nm?

(III) Suppose two stars of the same apparent brightness b are also believed to be the same size. The spectrum of one star peaks at 750 nm whereas that of the other peaks at 450 nm. Use Wien’s law and the Stefan-Boltzmann equation (Eq. 19–17) to estimate their relative distances from us. [Hint: See Examples 44–4 and 44–5.]

(I) Calculate the wavelength at the peak of the blackbody radiation distribution at 2.7 K using Wien’s law.

(II) A high-energy pulsed laser emits a 1.0-ns-long pulse of average power 1.5 x 10¹¹ W. The beam is nearly a cylinder 1.8 x 10⁻³ m in radius. Determine (a) the energy delivered in each pulse, and (b) the rms value of the electric field.