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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 -Trepresents the momentum flux density.) How is the momentum flux density related to the energy density, in this case?

Short Answer

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Answer

The all the elements Tzzare 0, where Tzz-ε0E02cos2(kz-ωt+δ), the answer make sense as the direction of the field is in the z direction. The relation between momentum flux density and the energy is 1cuz^.

Step by step solution

01

Expression for the electric field, and magnetic field:

Write the expression for the electric field.

Ez,t=E0coskz-ωtx^ ……. (1)

Write the expression for the magnetic field.

Bz,t=1cE0coskz-ωty^ ……. (2)

02

Determine the required relation:

The momentum flux density Tijis given by,

Tij=ε0(EiEj-12δijE2)+1μ0(BiBj-12δijB2)

With the fields in Eq. 9.48, E has only an x component, and B only has a y component. So, all the “off-diagonal” (ij)terms will be zero.

As for the “diagonal” elements:

Txx=ε0(ExEx-12E2)+1μ0(-12B2)=12(ε0E2-1μ0B2)=0

Solve for second diagonal element.

Tyy=ε0(-12E2)+1μ0(ByBy-12B2)=12(-ε0E2+1μ0B2)=0

Solve for third diagonal element.

Tu=ε0(-12E2)+1μ0(-12B2)=-u

So, Tzz=-ε0E02cos2(kz-ωt+δ)(all other elements zero).

The momentum of these fields is in the z direction, and it is being transported in the z direction, so yes, it does make sense that Tzzshould be the only nonzero element in Tij.

It is known that localid="1658405424229" -T·dais the rate at which momentum crosses an area da. Here we have no momentum crossing areas oriented in the x or y-direction.

The momentum per unit time per unit area flowing across a surface oriented in thez-direction is,

-Tzz=u=gc

Therefore,

p=gcATpT=gcA=momentumperunittimecrossingareaA

It is known that momentum flux density is equal to energy density. Therefore,

g=1cε0E02cos2(kz-ωt+δ)z^=1cuz^

Therefore, the all the elements Tzzare 0, where Tzz-ε0E02cos2(kz-ωt+δ), the answer make sense as the direction of the field is in the z direction. The relation between momentum flux density and the energy is 1cuz^.

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Most popular questions from this chapter

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