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In writing Eqs. 9.76 and 9.77, I tacitly assumed that the reflected and transmitted waves have the same polarization as the incident wave—along the x direction. Prove that this must be so. [Hint: Let the polarization vectors of the transmitted and reflected waves be

n^T=cosθTx^+sinθTy^,n^R=cosθRx^+sinθRy^prove from the boundary conditions that θT=θR=0.]

Short Answer

Expert verified

It is proved that θR=θT=0.

Step by step solution

01

Expression for the reflection and transmission at normal incidence:

Let the xy plane form a boundary between the two linear media. A plane wave of frequency traveling in the z-direction and polarized in the x-direction.

Write the expression for reflected wave.

E~R(z,t)=E~0Rei(k1z-ωt)x^B~R(z,t)=1v1E~0Rei(k1z-ωt)y^

Write the expression for the transmitted wave.

localid="1657519446367" E~T(z,t)=E~0Tei(k2z-ωt)x^B~T(z,t)=1v2E~0Tei(k2z-ωt)y^

02

Prove θT=θR=0:

Using a boundary condition,

E1''=E2''E~01+E~0R=E~0T...........(1)

Again use boundary condition,

1μ1B1=1μ2B2E~01-E~0R=βE~0T.............(2)

Hence, equation (1) is replaced as:

E~o1x^+E~0Rn^R=E~oTn^T ............(3)

Similarly, equation (2) is replaced as:

E~01y^-E~0R(z^×n^R)=βE~0T(z^×n^T)........(4)

Substitute the known values in equation (3).

E~01x^+E~0R(cosθRx^+sinθRy^)=E~0T(cosθTx^+sinθTy^) ….. (5)

Substitute the known values in equation (4).

E~01y^-E~0R(z^×cosθRx^+sinθRy^)=βE~0T(z^×cosθTx^+sinθTy^)E~01y^-E~0R(cosθRy^-sinθRx^)=βE~0T(cosθTy^+sinθTx^) ….. (6)

Write the x component from equation (6).

E~0RsinθR=-E~0TsinθT

Write the y-component from equation (5).

E~0RsinθR=E~0TsinθT

Hence, the above two equation can be satisfied only when.

Then, it is proved that,

θR=θT=0

Therefore, it is proved that θR=θT=0.

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Most popular questions from this chapter

Suppose

E(r,θ,ϕ,t)=Asinθr[cos(krωt)(1/kr)sin(krωt)]ϕ^

(This is, incidentally, the simplest possible spherical wave. For notational convenience, let role="math" localid="1658817164296" (krωt)u in your calculations.)

(a) Show that E obeys all four of Maxwell's equations, in vacuum, and find the associated magnetic field.

(b) Calculate the Poynting vector. Average S over a full cycle to get the intensity vector I. (Does it point in the expected direction? Does it fall off liker2, as it should?)

(c) Integrate role="math" localid="1658817283737" Ida over a spherical surface to determine the total power radiated. [Answer: 4πA2/3μ0c ]

(a) Show directly that Eqs. 9.197 satisfy Maxwell’s equations (Eq. 9.177) and the boundary conditions (Eq. 9.175).

(b) Find the charge density, λ(z,t), and the current,I(z,t) , on the inner conductor.

The intensity of sunlight hitting the earth is about 1300Wm2 . 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?

Show that the modeTE00 cannot occur in a rectangular wave guide. [Hint: In this caseωc=k , so Eqs. 9.180 are indeterminate, and you must go back to Eq. 9.179. Show that is a constant, and hence—applying Faraday’s law in integral form to a cross section—thatBz=0 , so this would be a TEM mode.]

A microwave antenna radiating at 10GHzis to be protected from the environment by a plastic shield of dielectric constant2.5. . What is the minimum thickness of this shielding that will allow perfect transmission (assuming normal incidence)? [Hint: Use Eq. 9.199.]

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