Transverse magnetic waves in parallel planes

DEFINITION TRANSVERSE MAGNETIC (TM) WAVES OR E WAVES   :

In this case, the component of magnetic field vector H lies in the plane transverse to the direction of propagation that is there is no component of H along the direction of propagation where as component of electric field vector E lies along the direction of propagation.

Derivation transverse magnetic waves between parallel planes:

As the direction of propagation is assumed as z-direction, therefore,

Hz = 0, Ez not equals to 0

By substituting Hz = 0 in equation, we get

Hx = 0, Ey = 0,and Ex not equals to 0, Hy not equals to 0

Now write wave equation for free space in term of H

Ñ2H =d

Ñ2H =w2μεH                           [.  . =g2g(jwμ) (σ +jwε)

As σ = 0, then g2g=-w2με]

For the y-component, wave equation will become

d2Hg/dx2 + d2Hg/dy2 + d2Hg/dz2 = – w2μεHy

As      dHy/dy = 0 and d2Hg/dz2 =  g2gHy               [using assumptions (e) and (f) of article “waves between parallel planes“]

The wave equation becomes

d2Hg/dx2 + g2gHy = – w2μεHy

or                   d2Hg/dx2 = -(g2g + w2με) Hy

or                   d2Hg/dx2 = – K2g Hy                                                                ..(14) (The equation number is continued from article of “transverse electric waves“)

where                       K2g = Y2g + w2με

As                  Hy = Hy0  e ggz                                                                                    ..(15(a))

d2Hg/dx2 =( d2Hg0/dx2 ) e ggz                                                           ..(15(b))

By substituting equations (15) in equation (14), wave equation becomes

d2Hg0/dx2 = – K2g Hy0

The above equation is a standard differential equation of simple harmonic motion and its solution can be written in the form

Hg0 = A3 sin Kgx + A4 cos Kgx

Hg =( A3 sin Kgx + A4 cos Kgx) e ggz                         ..(16)

(because Hg = Hg0 e ggz)

where A3 and A4 are arbitrary constants.

Here, the boundary conditions cannot be applied directly to Hg to determine the constants  A3 and A4 because, the tangential component of H is not zero at the surface of a conductor (Htan ¹ 0). However from equation Ez will be obtained in terms of Hg , and then the boundary conditions would be applied to  Ez

Differentiate equation w.r.t.  x

dHg/dx = Kg (A3 cos Kgx – A4 sin Kgx) e ggz

From equation

Ez = (1/ jwε) dHy/dx

Put value of dHg/dx in above equation

Ez =( Kg/ jwε) [A3 cos Kgx – A4 sin Kgx) e ggz                                 ..(17)

Applying the boundary condition

that  Ez = 0 at x = 0 in equation, we get

0 =( Kg/ jwε) [A3 cos Kg0 – A4 sin Kgx0) e ggz

Equation reduces to

Ez =( Kg/ jwε) [ – A4 sin Kgx) e ggz                                                    ..(18)

Now applying boundary condition

that  Ez = 0 at x = a in equation, we get

0 =( Kg/ jwε) [- A4 sin Kgα) e ggz

Or                              sin Kgα = 0

Or                              Kga = mπ

Or                              Kg = mπ/α

Equation becomes

Ez =( -Kg A4/ jwε) [ sin (mπ x/α)] e ggz

Ez = mπ x / jwεα   A4 sin (mπ x/α)] e ggz                                                  (gg = jbg)          ..(19(a))

Substitute equation(19(a)) in 6(c) and integrate

(dHy/dg)dx = jwε (=KgA4/ jwε) ( [ sin (mπ x/α)] e ggz)dx

Or                  Hy = A4 (mπ /α) (a/mπ ) cos (mπx /α) e ggz

Or                  Hy = A4 cos (mπx /α) e-jbgz                                                               ..(19(b))

From              Ex = ggHy

Put equation in above equation

Ex = (gg/ jwε) A4 cos (mπ x/α) e ggz

Ex = (bg/ wε) A4 cos (mπ x/α) e -jbgz                                                           ..(19(c))

Equations (19(a),(b),(c)) represent the expressions for field component for TM waves. In the case of Transverse Magnetic waves there is also the probability of m = 0, as by putting m = 0, in equations (19), some of the fields exist (e.g. Ex and Hy. Therefore the lowest order mode that can exist in TM waves in the TM0 mode.

This is the definition, derivation and discussion of transverse magnetic (TM) waves between parallel planes.

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