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Consider the resonant cavity produced by closing off the two ends of a rectangular wave guide, at z=0 and at z=d, making a perfectly conducting empty box. Show that the resonant frequencies for both TE and TM modes are given by

ωlmn=(ld)2+(ma)2+(nb)2 (9.204)

For integers l, m, and n. Find the associated electric and magneticfields.

Short Answer

Expert verified

The resonant frequencies for both TE and TM modes are ωlmn=ld2+ma2+nb2, the associated electric field is Ex=Bcoskxxsinkzzx^+Dsinkxxcoskyysinkzzy^+Fsinkxxsinkyycoskzzz^, and the associated magnetic field is Bx=-iωFky-Dkzsinkxxcoskyycoskzzx^-iωBkz-Fkxcoskxxsinkyycoskzzy^-iωDkx-Bkycoskxxcoskyysinkzzz^.

Step by step solution

01

Expression for the resonant frequencies for both TE and TM mode:

Write the expression for the resonant frequencies for both TE and TM mode.

ω2=c2(kx2+ky2+kz2) …… (1)

Here, c is the speed of light and k is the wave number.

Here, the value of data-custom-editor="chemistry" kx,data-custom-editor="chemistry" ky anddata-custom-editor="chemistry" kz are given as:

kx=aky=bkz=d

02

Prove the expression for resonant frequencies for both TE and TM mode:

Substitute kx=a,ky=b andkz=d in equation (1).

ω2=c2a2+b2+d2ω=ma2+nb2+ld2

03

Determine the associated electric field:

Write the expression for the x, y, and z components of an electric field.

Exx,y,z=Asinkxx+BcoskxxsinkxysinkzzEyx,y,z=sinkxxCsinkyy+DcoskyysinkzzEzx,y,z=sinkxxsinkyyEsinkzz+Fcoskzz.

Hence, the associated electric field will be,

role="math" localid="1657688765767" Ex=Bcoskxxsinkzzx^+Dsinkxxcoskyysinkzzy^+Fsinkxxsinkyycoskzzz^

04

Determine the associated magnetic field:

Write the expression for the x component of a magnetic field.

Bx=-iωEzy-Eyz

Substitute the value ofEz andEy in the above expression.

Bx=-iωFkysinkxxcoskyycoskzz-Dkzsinkxxcoskyycoskzz

Write the expression for the y component of a magnetic field.

By=-iωExz-Ezx

Substitute the value ofdata-custom-editor="chemistry" Ex anddata-custom-editor="chemistry" Ez in the above expression.

data-custom-editor="chemistry" By=-iωBkzcoskxxsinkyycoskzz-Fkxcoskxxsinkyycoskzz

Write the expression for the z component of a magnetic field.

data-custom-editor="chemistry" Bz=-iωEx-Exy

Substitute the value ofdata-custom-editor="chemistry" Ey anddata-custom-editor="chemistry" Ex in the above expression.

data-custom-editor="chemistry" Bz=-iωDkxcoskxxcoskyysinkzz-Bkycoskxxcoskyysinkzz

Hence, the associated magnetic field will be,

data-custom-editor="chemistry" B=-iωFky-Dkzsinkxxcoskyycoskzzx^-iωBkz-Fkxcoskxxsinkyycoskzzy^-iωDkx-Bkycoskxxcoskyysinkzzz^

Therefore, the resonant frequencies for both TE and TM modes are ω=cπma2+nb2+ld2, the associated electric field is Ex=Bcoskxxsinkzzx^+Dsinkxxcoskyysinkzzy^+Fsinkxxsinkyycoskzzz^, and the associated magnetic field is data-custom-editor="chemistry" B=-iωFky-Dkzsinkxxcoskyycoskzzx^-iωBkz-Fkxcoskxxsinkyycoskzzy^-iωDkx-Bkycoskxxcoskyysinkzzz^.

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