Electromagnetism, Science

Transverse electric waves


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

Derivation of transverse electric waves in parallel planes:

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

Ez = 0 and Hz is not equal to 0

Now by substituting Ez = 0 in equation (8) of article “waves between parallel planes”, we get

Ex= 0 and Hy = 0 and

Ey not equals to 0 , H not equals to 0

Now write wave equations for free space in terms of E

Ñ2E =g2gE

= -w2μεE                     (because g2g = (jwμ) (σ + jwε) As σ =0(from                                                                            assumption  (c) of article “waves between parallel planes” => g2g =-w2με)

Or                    d2E/dx2 + d2E/dy2 + d2E/dz2 = -w2μεE

For the y component, the wave equation will become

d2Ey/dx2 + d2Ey/dg 2 + d2Ey/dz2 = -w2μεEy

also                       dEg/dg  = 0                                                      [using assumption (e)]

and                       Eg = Eg0 e ggz                                                                    [using assumption (f)] of article “waves between parallel planes”

Thus           d2Eg/dz2 = g2Eg2gEg2gEg

By substituting above values of dEg/dg and d2Ey/dz2 in wave equation, we get

d2Eg/dx2  + g2gEg= -w2μεEg

or              d2Eg/dx2  = -(g2g + w2με) Eg

or              d2Eg/dx2  = –K2g Eg                                                                  ..(9)

where      K2g = g2g + w2με

as              Eg = Eg0  e ggz                                                                          ..(10(a))

thus           d2Eg/dx2  = d2Eg0 e ggz/dx2                                                                    ..(10(b))

by substituting equation(10) in equation (9) , wave equation becomes

d2Eg0/ dx2 = K2g Eg0 

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

Eg0 = A1 sin Kgx + A2cos Kgx

Or              Eg = (A1 sin Kgx + A2 cos Kgx) e ggz                                      ..(11)

Where A1 and A2 are arbitrary constants.

A1 and A2 can be determined with the help of following boundary conditions :

Etan = 0 at the surface of conductor

This implies that

                              Ey = 0 at x = 0

Ey = 0 at x = a

Thus apply boundary condition

That           Ey = 0 at x = 0

So equation (11)will become

0 = (A1 sin Kg0 + A2 cos Kg0) e ggz

Thus equation (11) reduces to

Ey = A1 sin Kgxe ggz                                                   ..(12)

Now apply boundary condition

That Ey = 0 at x = a in equation, we get

0 = A1 sin Kgae ggz

Or              sinKga = 0 as A1¹ 0

Or              Kga = mπ

Or              Kg    = mπ/a, wher m = 1,2,3

(If m = 0, all field components vanish. Its a special case and will be discussed later)

.           equation (12)becomes

. .         Ey = A1 sin (mπ/a)e ggz

Ey = A1 sin (mπ/a)ejbgz       (. . gg =  jbg  if αg = 0)             ..(13(a))

Put equation  (13 a)in (8 d of article “waves between parallel planes”)and integrate

A1 sin (mπx/a)e ggz = jwμ/ K2g dHz/dx   dx

Hz = A1 K2g e ggz /jwμ          sin(mπx/a)/dx

= – A1m2π2/a2   e ggz/ jwμ  cos (mπx/a)/ mπ      where K2g = m2π2/a2

Hz =- A1 mπ cos (mπx/a) e ggz/ jwμ

or                Hz =- A1 mπ cos (mπx/a) e ggz/ jwμ                       (gg = jbg)          ..(13(b))

Differentiate (13(b))w.r.t.  x

dHz/dx  = A1m2π2 sin (mπx/a) e ggz/ jwμa2

Substitute above equation in equation

Hx=- gg / K2g A1 m2π2/ jwμa2 sin (mπx/a) e ggz

Hx=- gg m2π2/a2         A1 m2π2/jwμa2     sin (mπx/a) e ggz

Hx=- gg  A1 /jwμ   sin (mπx/a) e ggz

Hx=- bg  A1 /wμ   sin (mπx/a) e ggz                 ( gg =  jbg)        ..(13(c))

Equation (13(a)), (b), (c)) represent the expressions for field components for TE waves. Each value of m in equations (13) represent a particular field configuration and the wave associated with integer m is designated as TEm wave. The lowest order mode that can exist in this case is TE1 mode.

This is the definition, discussion and derivation of transverse electric (TE) waves between parallel planes or plates.


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