# Waves between parallel planes

Let us discuss how waves propagate through parallel planes and derive the necessary relation of transverse electric and magnetic waves:

**Assumptions :**

(a) Pair of parallel planes are perfectly conducting.

(b) Separation between the planes is ‘a’ meter in x – direction.

(c) Space between planes is perfect dielectric (σ = 0) of permittivity ε and permeability μ.

(d) Planes are of infinite extent in the y and z direction.

(e) As the plane is extended to infinity in the y – direction there are no boundary conditions to be met in this direction, therefore field is uniform in the y- direction i.e. derivative with respect to y is zero (d/dy =0)

(f) Direction of propagation of wave is along z-direction, therefore the variation of all the field component in the z-direction is expressed as e^{-y}_{g}^{z}

Where y_{g} = a_{g} + jb_{g}

Here y_{g}is propagation constant and it is not equal to y(y_{g }¹ y). In special case of uniform plane waves, y_{g} reduces to y.

a_{g} is attenuation constant, and

b_{g} is phase constant.

(g) In time varying form, the field variation is expressed as

e^{jwt} e^{-y}_{g}^{z} = e^{ (jwt – y}_{g}^{z)}

^{ }e^{(jwt – (a}_{g }+ ^{j}b_{g}^{)z)}

If there is no attenuation, a_{g} = 0 then field variation is expressed as

^{ }e^{jwt} ^{- j}b_{g}^{z} = e^{ j(wt – }b_{g}^{z)}

**Boundary Conditions :**

In order to determine the electromagnetic field configuration between parallel planes, Maxwell’s field equation are solved with the following boundary condition :

(I) Electric field must terminate normally on the conductor, that is, tangential component of electric field must be zero.

E_{tan} = 0

(II) Magnetic field must lie tangentially along the wall surface, that is, the normal component of magnetic field must be zero.

H_{nor} = 0

*Derivation of field equations :*

In general, Maxwell’s equations (Modified Ampere’s Circuital law and Faraday’s law of em induction) in non-conducting region (σ = 0) between the planes are

Ñ x **H** = jw**εE** (Modified Ampere’s circuital law) (1)

Ñ x **E** = jwμ**H** (Faraday’s law of em iduction) (2)

Expanding equation (1) in rectangular coordinates, we get

**a _{x } a_{y }a_{z}**

Ñ x **H** = d/dx d/dy d/dz = jw**ε** (E_{x}a_{x} + E_{y}a_{y }+ E_{z}a_{z})

**H**_{X }**H**_{Y }**H**_{Z}

** a**_{x} d**H**_{z}/dg – d**H**_{y}/dz – **a**_{y} d**H**_{z}/dx– d**H**_{x}/dz + **a**_{y} d**H**_{y}/dx– d**H**_{x}/dg

** = j**wε**E _{x}a**

_{x }+

**j**wε

**E**

_{y}a_{y}+

**j**wε

**E**

_{z}a_{z}

Comparing the respective components on both sides, we get

dH_{z}/dg - dH_{y}/dz = **j**wε**E _{x}**

dH_{x}/dz - dH_{z}/dx = **j**wε**E _{y}**

dH_{y}/dx - dH_{x}/dg = **j**wε**E _{z }(3)**

Similarly expanding equation (2) and equating respective components on both sides, we get

d**E**_{z}/dg - d**E**_{y}/dz = **j**wμ**H _{x}**

d**E**_{x}/dz - d**E**_{z}/dx = **j**wμ**H _{y}**

d**E**_{y}/dx - d**E**_{x}/dg = **j**wμ**H _{z }(4)**

From assumption (f), as the direction of propagation is along z-direction, the variation of field components can be expressed as

**H**_{x} = **H**_{x0} e^{-y}_{g}^{z}

Thus d**H**_{x} /dz =^{-y}_{g}H_{x0} e^{-y}_{g}^{z}

Or d**H**_{x} /dz = -g_{g}**H**_{x }(5a)

Similarly dH_{y}/dz = -g_{g}H_{y }(5b)

dE_{x}/dz = -g_{g}E_{x }(5c)

and dE_{y}/dz = -g_{g}E_{y }(5d)

also from assumption (e), d/dy = 0

dH_{z}/dg - dH_{x}/dg = d**E**_{z}/dg = d**E**_{x}/dg = 0 (5e)

By substituting equations 5a, b and e, we have

g_{g}H_{y} = **j**wε**E _{x }**(6a)

-g_{g}H_{x} – dH_{z}/dx = **j**wε**E _{y }**(6b)

** **d**H _{y}**/dx =

**j**wε

**E**(6c)

_{z }

_{ }Similarly by substituting equations 5c, d and e in equation 4, we have

g_{g}E_{y} = -** j**wμ**H _{x }**(7a)

** _{-}**g

_{g}E

_{x}-dE

_{z}/dx = -jwμ

**H**(7b)

_{y }dE_{y}/dx = -** j**wμ**H _{z }**(7c)

**Now use equations 6a and 7b**

From equation 6a

Ex = g_{g}H_{y}/jwε

Putting value of Ex in equation 7(b), we have

g^{2}_{g}H_{y}/jwε + dEz/dx = jwμ**H _{y}**

** **

dE_{z}/dx =( **j**wμ – g^{2}g/**j**wε)H_{y}

**j**wε (dE_{z}/dx) = (-w^{2}με – g^{2}g) H_{y}

**j**wε (dE_{z}/dx) = -H_{y}(g^{2}g+ H_{y} w^{2}με)

= – H_{y}K^{2}g

K^{2}g = g^{2}g + w^{2}με

H_{y} = -** j**wε/ K^{2}g dE_{z}/dx (8a)

**Again use equations 6a and 7b**

From equation

H_{y} = 1/** j**wμ ( dE_{z}/dx + g_{g}E_{x})

Substituting value of H_{y} in equation , we have

g_{g}/** j**wμ ( dE_{z}/dx + g^{2}gEx/jwμ )= jwE_{x}

g_{g}/** j**wμ ( dE_{z}/dx ) (jwε- g2g/** j**wμ)E_{x}

g_{g}(dE_{z}/d_{x}) = (-w^{2} με – g2g) E_{x}

_{ -} Y_{g}(dE_{z}/d_{x}) = E_{x}K^{2}_{g}

K^{2}_{g} = g^{2}g + w^{2} με

E_{x} = (gg/ K^{2}_{g}) dE_{z}/dx (8b)

**Similarly by using and solving equations 6b and 7a, we get**

H_{x} = (-gg/ K^{2}_{g}) dH_{z}/dx 8c

and E_{y}= (jwμ/ K^{2}_{g}) dH_{z}/dx 8d

where K^{2}_{g} = g^{2}_{g} + w^{2}με

Equations 8(a,b,c and d) represent the equations of plane waves propagating in +z direction varying sinusoidally between the infinite parallel planes.

In equation, the components of electric and magnetic fields strengths are expressed in terms of E_{z} and H_{z}.

If E_{z} = 0 and H_{z} = 0, all the components will vanish, therefore it is observed that there must be a z component of either E or H i.e. comonent along the direction of propagation.

Therefore, the propagating waves in parallel plane guide are classified into following types according to whether E_{z} or H_{z }exists :

- Transverse Electric (TE) Waves or H Waves (E
_{z}= 0, H_{z}¹ 0) - Transverse Magnetic (TM) Waves or E Waves (H
_{z}= 0, E_{z}¹ 0)

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