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Chapter 9: Q12P (page 400)

In the complex notation there is a clever device for finding the time average of a product. Suppose f(r,t)=Acos(k脳r-蝇t+未a)and g(r,t)=Bcos(k脳r-蝇t+未b). Show that <fg>=(1/2)Re(fg~), where the star denotes complex conjugation. [Note that this only works if the two waves have the same k and蝇, but they need not have the same amplitude or phase.] For example,

<u>=14Re(蔚0E~脳E~+1渭0B~脳B~)and<S>=12渭0Re(E~脳B).~ and .

Short Answer

Expert verified

It is proved that fg=1/2Ref~g~.

Step by step solution

01

Expression for the f(r,t) and g(r,t):

Write the expression for f (r , t).

f(r,t)=Acos(k.r-蝇t+未a) 鈥�. (1)

Write the expression for g( r , t).

g(r,t)=Bcos(k.r-蝇t+未b) 鈥�. (1)

02

Determine the <fg>:

Find the fgas follows.

fg=1T鈭�gTAcos(k.r-蝇t+未a).Bcos(k.r-蝇t+未b)dt=ABT鈭�gTcos(2k.r-蝇t+未a+未b+cos(未a-未b)dt=ABTcos(未a-未b)T=12ABcos(未a-未b)

03

Determine (12)Re(fg)^:

Write the equation in the complex notation.

f~=A~ei(k.r-蝇t)g~=B~e-i(k.r-蝇t)

Where, A~=aei未aandB~=Be-i未b.

Determine12f~g*~ as follows.

12f~g*~=A~ei(k.r-蝇t)B~ei(k.r-蝇t)=12A~B~=12ABei(未a-未b)=12AB(cos(未a-未b)+isin(未a-未b))

Consider the real term.

Re12(f~g~)=12ABcos(未a-未b)=fg

Therefore, it is proved that fg=(1/2)Re(f~g~).

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Most popular questions from this chapter

Question:Equation 9.36 describes the most general linearly polarized wave on a string. Linear (or "plane") polarization (so called because the displacement is parallel to a fixed vector n) results from the combination of horizontally and vertically polarized waves of the same phase (Eq. 9.39). If the two components are of equal amplitude, but out of phase by (say,未谓=0,未h=90掳,), the result is a circularly polarized wave. In that case:

(a) At a fixed point, show that the string moves in a circle about the axis. Does it go clockwise or counter clockwise, as you look down the axis toward the origin? How would you construct a wave circling the other way? (In optics, the clockwise case is called right circular polarization, and the counter clockwise, left circular polarization.)

(b) Sketch the string at time t =0.

(c) How would you shake the string in order to produce a circularly polarized wave?

[The naive explanation for the pressure of light offered in Section 9.2.3 has its flaws, as you discovered if you worked Problem 9.11. Here鈥檚 another account, due originally to Planck.] A plane wave traveling through vacuum in the z direction encounters a perfect conductor occupying the region z鈮�0, and reflects back:

E(z,t)=E0[cos(kz-蝇t)-cos(kz+蝇t)]x^,(z>0),

(a) Find the accompanying magnetic field (in the region role="math" localid="1657454664985" (z>0).

(b) Assuming inside the conductor, find the current K on the surface z=0, by invoking the appropriate boundary condition.

(c) Find the magnetic force per unit area on the surface, and compare its time average with the expected radiation pressure (Eq. 9.64).

Find the width of the anomalous dispersion region for the case of a single resonance at frequency 蝇0. Assume纬<<蝇0 . Show that the index of refraction assumes its maximum and minimum values at points where the absorption coefficient is at half-maximum.

(a) Show that the skin depth in a poor conductor 蟽<<蝇蔚is (蔚蟽)2渭(independent of frequency). Find the skin depth (in meters) for (pure) water. (Use the static values of 蔚,渭and 蟽; your answers will be valid, then, only at relatively low frequencies.)

(b) Show that the skin depth in a good conductor (蟽<<蝇蔚)is 位2蟺(where 位 is the wavelength in the conductor). Find the skin depth (in nanometers) for a typical metal (蟽>>惟m107-1)in the visible range (蝇鈮�1015/s), assuming 蔚=蔚0and 渭鈮�渭0. Why are metals opaque?

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

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

鈭�Fdrag=-Y鈭�f鈭�t鈭�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 f~(z,t)=ei蝇tF~(z).

(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 违,T,渭and 蝇.

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