魅影直播

(a) Formulate an appropriate boundary condition, to replace Eq. 9.27, for the case of two strings under tension T joined by a knot of mass m.

(b) Find the amplitude and phase of the reflected and transmitted waves for the case where the knot has a mass m and the second string is massless.

Suppose string 2 is embedded in a viscous medium (such as molasses), which imposes a drag force that is proportional to its (transverse) speed:

$\mathbf{鈭�}{\mathit{F}}_{\mathbf{d}\mathbf{r}\mathbf{a}\mathbf{g}}\mathbf{=}\mathbf{-}\mathit{Y}\frac{\mathbf{鈭�}\mathbf{f}}{\mathbf{鈭�}\mathbf{t}}\mathbf{鈭�}\mathit{z}$.

(a) Derive the modified wave equation describing the motion of the string.

(b) Solve this equation, assuming the string vibrates at the incident frequency. That is, look for solutions of the form $\stackrel{\mathbf{~}}{\mathbf{f}}\left(z,t\right)\mathbf{=}{\mathit{e}}^{\mathbf{i}\mathbf{蝇}\mathbf{t}}\stackrel{\mathbf{~}}{\mathbf{F}}\left(z\right)$.

(c) Show that the waves are attenuated (that is, their amplitude decreases with increasing z). Find the characteristic penetration distance, at which the amplitude is of its original value, in terms of $\mathit{违}\mathbf{,}\mathit{T}\mathbf{,}\mathit{渭}$and $\mathit{蝇}$.

(d) If a wave of amplitude A , phase $\mathit{未}$,= 0 and frequency$\mathit{蝇}$ is incident from the left (string 1), find the reflected wave鈥檚 amplitude and phase.

The intensity of sunlight hitting the earth is about $\mathbf{1300}\frac{\mathbf{W}}{{\mathbf{m}}^{\mathbf{2}}}$ . If sunlight strikes a perfect absorber, what pressure does it exert? How about a perfect reflector? What fraction of atmospheric pressure does this amount to?

Consider a particle of charge q and mass m, free to move in the $\mathit{x}\mathit{y}$plane in response to an electromagnetic wave propagating in the z direction (Eq. 9.48鈥攎ight as well set $\mathit{未}\mathbf{=}\mathbf{0}\mathbf{)}$).

(a) Ignoring the magnetic force, find the velocity of the particle, as a function of time. (Assume the average velocity is zero.)

(b) Now calculate the resulting magnetic force on the particle.

(c) Show that the (time) average magnetic force is zero.

The problem with this naive model for the pressure of light is that the velocity is ${\mathbf{90}}^{\mathbf{掳}}$out of phase with the fields. For energy to be absorbed there鈥檚 got to be some resistance to the motion of the charges. Suppose we include a force of the form $\mathit{y}\mathit{m}\mathit{v}$, for some damping constant $\mathit{y}$.

(d) Repeat part (a) (ignore the exponentially damped transient). Repeat part (b), and find the average magnetic force on the particle.

Find all elements of the Maxwell stress tensor for a monochromatic plane wave traveling in the z direction and linearly polarized in the x direction (Eq. 9.48). Does your answer make sense? (Remember that $\mathbf{-}\stackrel{\mathbf{鈫�}}{\mathbf{T}}$represents the momentum flux density.) How is the momentum flux density related to the energy density, in this case?

Calculate the exact reflection and transmission coefficients, without assuming ${渭}_1={渭}_2={渭}_0$. Confirm that R + T = 1.

Question:The index of refraction of diamond is 2.42. Construct the graph analogous to Fig. 9.16 for the air/diamond interface. (Assume .) In particular, calculate (a) the amplitudes at normal incidence, (b) Brewster's angle, and (c) the "crossover" angle, at which the reflected and transmitted amplitudes are equal.

(a) Suppose you imbedded some free charge in a piece of glass. About how long would it take for the charge to flow to the surface?

(b) Silver is an excellent conductor, but it鈥檚 expensive. Suppose you were designing a microwave experiment to operate at a frequency of${\mathbf{10}}^{\mathbf{10}}\mathbf{}\mathit{H}\mathit{z}$. How thick would you make the silver coatings?

(c) Find the wavelength and propagation speed in copper for radio waves at role="math" localid="1655716459863" $\mathbf{1}\mathbf{}\mathit{M}\mathit{H}\mathit{z}$. Compare the corresponding values in air (or vacuum).

By explicit differentiation, check that the functions ${\mathit{f}}_{\mathbf{1}}$, ${\mathit{f}}_{\mathbf{2}}$, and ${\mathit{f}}_{\mathbf{3}}$in the text satisfy the wave equation. Show that ${\mathit{f}}_{\mathbf{4}}$and ${\mathit{f}}_{\mathbf{5}}$do not.

(a) Show that the skin depth in a poor conductor $\mathit{蟽}\mathbf{<}\mathbf{<}\mathit{蝇}\mathit{蔚}$is $\left(\frac{蔚}{蟽}\right)\sqrt{\frac{\mathbf{2}}{\mathbf{渭}}}$(independent of frequency). Find the skin depth (in meters) for (pure) water. (Use the static values of $\mathit{蔚}\mathbf{}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\mathit{渭}$and $\mathit{蟽}$; your answers will be valid, then, only at relatively low frequencies.)

(b) Show that the skin depth in a good conductor $\left(蟽<<蝇蔚\right)$is $\frac{\mathbf{位}}{\mathbf{2}\mathbf{蟺}}$(where 位 is the wavelength in the conductor). Find the skin depth (in nanometers) for a typical metal $\left(蟽>>惟{\left(m{10}^7\right)}^{-1}\right)$in the visible range $\left(蝇鈮�{10}^{15}/s\right)$, assuming $\mathit{蔚}\mathbf{=}{\mathit{蔚}}_{\mathbf{0}}$and $\mathit{渭}\mathbf{鈮�}{\mathit{渭}}_{\mathbf{0}}$. Why are metals opaque?

(c) Show that in a good conductor the magnetic field lags the electric field by ${\mathbf{45}}^{\mathbf{掳}}$, and find the ratio of their amplitudes. For a numerical example, use the 鈥渢ypical metal鈥� in part (b).