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(a) Calculate the (time-averaged) energy density of an electromagnetic plane wave in a conducting medium (Eq. 9.138). Show that the magnetic contribution always dominates.

(b) Show that the intensity is(k2ฮผฯ‰)E02e-2xz

Short Answer

Expert verified

(a) The energy density of an electromagnetic plane wave in a conductive medium isu=k22ฮผฯ‰2E02eโˆ’2kz .

(b) It is proved that the intensity is (k2ฮผฯ‰)E02e-2xz.

Step by step solution

01

Expression for the energy density of an electromagnetic plane wave:

Write the expression for the energy density of an electromagnetic plane wave.

u=12(ฮตE2+1ฮผB2) โ€ฆโ€ฆ (1)

Here,ฮต is the permittivity of free space, E is the electric field, ฮผIs the permeability of free space, and B is the magnetic field.

02

Determine the energy density of an electromagnetic plane wave in a conductive medium:

(a)

Write the expression for the electric field.

E~(z,t)=E0eโˆ’kzcos(kzโˆ’ฯ‰t+ฮดE)x^

Squaring on both sides.

E2=E02eโˆ’2kzcos2(kzโˆ’ฯ‰t+ฮดE)

Write the expression for the magnetic field.

B~(z,t)=B0eโˆ’kzcos(kzโˆ’ฯ‰t+ฮดE+ฯ•)y^

Squaring on both sides.

B2=B02eโˆ’2kzcos2(kzโˆ’ฯ‰t+ฮดE+ฯ•)

SubstituteE2=E02eโˆ’2kzcos2(kzโˆ’ฯ‰t+ฮดE) andB2=B02eโˆ’2kzcos2(kzโˆ’ฯ‰t+ฮดE+ฯ•) in equation (1).

u=12(ฮตE02eโˆ’2kzcos2(kzโˆ’ฯ‰t+ฮดE)+1ฮผB02eโˆ’2kzcos2(kzโˆ’ฯ‰t+ฮดE+ฯ•))u=12eโˆ’2kz[ฮตE02cos2(kzโˆ’ฯ‰t+ฮดE)+B02ฮผcos2(kzโˆ’ฯ‰t+ฮดE+ฯ•)]u=12eโˆ’2kz[ฮต2E02+12ฮผB02]

Using the relation between B0andE0 , it is known that:

B0=E0ฮตฮผ1+(ฯƒฮตฯ‰)2

Calculate the energy density of an electromagnetic plane wave in a conductive medium.

u=12eโˆ’2kz[ฮต2E02+12ฮผ(E0ฮตฮผ1+(ฯƒฮตฯ‰)2)2]u=14eโˆ’2kzฮตE022ฮตฮผk2ฯ‰2u=k22ฮผฯ‰2E02eโˆ’2kz

03

Show that the magnetic contribution always dominates:

Write the expression for the magnetic energy.

umag=(B022ฮผ)12eโˆ’2kz

Write the expression for the density electrostatic energy density.

uelec=12eโˆ’2kzฮต2E02

Take the ratio of magnetic energy density and electrostatic energy density.

umaguelec=(B022ฮผ)12eโˆ’2kz12eโˆ’2kzฮต2E02umaguelec=(B02ฮผ)ฮตE02umaguelec=1ฮตฮผฮตฮผ1+(ฯƒฮตฯ‰)2

On further solving,

umaguelec=1+(ฯƒฮตฯ‰)2>1umag>uelec

Hence, the magnetic contribution is always the dominator.

Therefore, the energy density of an electromagnetic plane wave in a conductive medium isu=k22ฮผฯ‰2E02eโˆ’2kz .

04

Show that the intensity is (k2ฮผฯ‰)E02e-2xz(k2ฮผฯ‰)E02e-2xz:

(b)

Write the expression for intensity.

I=cu

Substituteu=k22ฮผฯ‰2E02eโˆ’2kz and c=ฯ‰kin the above expression.

I=k22ฮผฯ‰2E02eโˆ’2kzฯ‰kI=(ฯ‰k)k22ฮผฯ‰2E02eโˆ’2kzI=k2ฮผฯ‰E02eโˆ’2kz

Therefore, it is proved that I=k2ฮผฯ‰E02eโˆ’2kz.

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Most popular questions from this chapter

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

โˆ†Fdrag=-Yโˆ‚fโˆ‚tโˆ†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)=eiฯ‰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 ฮฅ,T,ฮผand ฯ‰.

(d) If a wave of amplitude A , phase ฮด,= 0 and frequencyฯ‰ is incident from the left (string 1), find the reflected waveโ€™s amplitude and phase.

The "inversion theorem" for Fourier transforms states that

ฯ•(Z)=โˆซ-โˆžโˆžฯ•(k)eikzdkโ‡”ฯ•(k)=12ฯ€โˆซ-โˆžโˆžฯ•(z)e-ikzdz

Use this to determine A(k), in Eq. 9.20, in terms of f(z,0)andf*(z,0)

Suppose

E(r,ฮธ,ฯ•,t)=Asinฮธr[cos(kr-ฯ‰t)-1krsin(kr-ฯ‰t)]ฯ•โœ

(This is, incidentally, the simplest possible spherical wave. For notational convenience, let(kr-ฯ‰t)โ‰กuin your calculations.)

(a) Show that Eobeys 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 . (Does it point in the expected direction? Does it fall off like r-2, as it should?)

(c) Integrate over a spherical surface to determine the total power radiated. [Answer:4ฯ€A2/3ฮผ0c]

By explicit differentiation, check that the functions f1, f2, and f3in the text satisfy the wave equation. Show that f4and f5do not.

Calculate the exact reflection and transmission coefficients, without assuming ฮผ1=ฮผ2=ฮผ0. Confirm that R + T = 1.

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