Definition of Electric line of force:

An electric line of force is a path, straight or curved, such that tangent to it at any point gives the direction of electric field intensity at that point.

In Fig. 1, PQ is an electrostatic line of force. The tangent to the line at any point A gives the direction of electric intensity  EA at A. Similarly, tangent to PQ at B gives the direction of EB, (see Figure)

Let us discuss the direction of electric field lines for different types of charges:

1. For a single positive point charge,   these lines are directed radially outwards.

The lines of force extend to infinity, (Fig. 2)

2. For a single negative point charge, these lines are directed radially inwards. (Fig. 3).

3. For pairs of equal and opposite charges,   lines of force are shown in Fig. 4.

Above is the definition of electric field lines and their directions for different types of charges. One can also understand magnetic flux similarly.

Properties Of Electric Lines Of Force :

1. Electric lines of force are discontinuous curves. They start from a positively charged body and end at a negatively charged body (Fig. 1.4). No electric lines of force exist inside the charged body.
2. Tangent to the line of force at any point gives the direction of electric intensity at that point.
3. No two electric lines of force can intersect each other. This is because at the point of intersection P, we can draw two tangents PA and PB to the two lines of force. This   would   mean  two   directions of electric intensity   at   the   same  point,   which is not possible. Hence no two lines of force can cross each other.

Above are the main properties of electric lines of force with the reason why no two electric lines of force can cross each other.

Following is our youtube video on Electric lines of force:

The post Electric Lines of Force or Field Lines and Properties first appeared on Winner Science.

This is so because invisible magnetic lines of forces emit through the magnet and if the iron piece is in range of magnet then it will be attracted towards the magnet otherwise not. These magnetic lines of force passing normally through a particular area is called magnetic flux.

Magnetic flux has units of webers. Stronger is the magnet, more will be the magnetic flux . It means more will be the power to attract. There is another concept of magnetic flux density. It is defined as the magnetic flux per unit area. Therefore it has the units of webers per square meter.

The symbol of magnetic flux is ø. Sometimes subscript m is used with this symbol. 

I hope you have understood the concept of magnetic flux and its units.

The post What is magnetic flux and its units first appeared on Winner Science.

Flux: Flux simply means the flow. For example if we say water flux, then it means flow of water through a particular area. Therefore let us now understand the meaning of electric flux.

Electric flux: Electric flux consists of two words “Electric” and “flux”. Electric comes from the word electric charges. When there will be electric charges then there will be electric field or electric field lines. “Flux” as I told earlier then it means Flow. Therefore, Electric flux means the flow of electric field lines from a particular area.

Let us say the small amount of Electric flux is dø and and Electric Field is E and it passes through a small area element dA. Then dø = E.dA. Here E and dA are vectors. As there is dot or scalar product, so the dø is scalar quantity. #

Thus dø = EdAcosθ

Units: The electric flux has SI units volt-meter. As electric field has units of volt/meter and area has square meter. Therefore dø that is electric flux has units of volt-meter.

I hope you have understood the meaning of flux and electric flux.

The post What is electric flux and its units first appeared on Winner Science.

Ans: We know that a charge at rest produces an electric field around it but no magnetic field where a moving charge produces both electric and magnetic fields but if the charge is moving with constant velocity there will be no charge in the values of the electric and magnetic fields,So all electromagnetic waves are produced.Now if charge is moving with non zero acceleration, both the electric and magnetic field will change, there by producing the em waves. So we conclude that an accelerated charge emits electromagnetic waves.

Important features or characteristics or facts about electromagnetic waves:

(i) em waves are produced by accelerated or oscillating charges.

(ii) They don’t require any material medium for their propagation.

(iii) They travel in free space with a speed of light.

(iv) A sinusoidal variation occurs in both the electric and magnetic field vectors.

(v) em waves are transverse in nature.

(vi) Velocity of em waves depends on electric and magnetic properties of medium through which travel and independent of amplitude of field vectors.

(vii) Velocity of em waves in a dielectric is less than velocity of light.

(viii) em waves carry energy which is distributed equally between electric and magnetic fields.

(ix) The electric vector is responsible for the optical effects of an em waves so it is known as the light vector.

(x) em waves are uncharged so they are not deflected by electric and magnetic fields.

 These are the features of electromagnetic waves.

The post Electromagnetic waves and their features first appeared on Winner Science.

(a) Cut – off Wavelength

ג_c = υ₀/f_c

Put the value of υ₀ and f_c in above equation,

Or                        ג_c = 1/√ μ₀ε₀   2 α√ μ₀ε₀/m

                 ג_c = 2 α/m

The above is the expression for the cut-off wavelength.

(b) Guide Wavelength

Definition. It is defined as the distance required for the phase to shift through 2π radians.

That is                  ג_g = 2 π/b_g

Or                                    ג_g = 2 π/ √ w²με – ( mπ /α)²

(because b_g = √ w²με – ( mπ /α)²)

This is the expression for  guide wavelength

Now at ¦_c ,                                 w²με = ( mπ /α)²

By substituting value of ( mπ /α)² in equation of ג_g , we get

ג_g = 2 π/ √ w²με – w²με

= 2 π / w√ με√(1 – w²_c / w² ) = 2 π / w√ μ⁰ε⁰ √(1 – w²_c / w² )

(if μ = μ₀ , ε=ε₀)

= 2 π υ₀/2 πf √(1 – w²_c / w² )

Or                        ג_g = ג/ √(1 – w²_c / w² )

Where ג = υ_0f is the free space wavelength.

As                        w_c = 2πf_c and w = 2πf

ג_g = ג/ √(1 – ¦_c² / f )

The above equation also represents guide wavelength in terms of free space and cut-off frequencies.

Derivation of the relation between ג ,ג_g and ג_c (or ג_g  in terms of ג ,ג_c) :

As                                    ג = ג/Ö(1 – ג/(גc)²

Therefore the expression of guide wavelength can also be written as

ג_g = ג/ Ö(1 – ג/(גc)²

This is the relation between ג_g, ג and ג_c.

Another  Form :

By squaring both sides of above equation, we get

ג²_g = ג²/1- ג/(גc)²

ג/(ג_g)² = 1- ג/(ג_c)²

Dividing both sides by ג², we get

1/ ג² = 1/ ג²_g +  ג²_c

This expression also represents the relation between ג ,ג_g and ג_c

(c) Phase (Wave) Velocity in parallel planes

Definition . It is the velocity of propagation of equiphase surface along the guide.

υ_p = w/b_g

Put the value of b_g in above expression

υ_p = w/√ w²με – ( mπ /α)²

when frequency becomes high enough so that w²με >> ( mπ /α)²

then                    υ_p => υ₀ = w/ √ w₂μ₀ε₀

or                         υ₀ =1/ √μ₀ε₀ = 3 * 10⁸ m/s

Now at ¦_c,                      w²_cμ₀ε₀ = ( mπ /α)²                           (if μ= μ₀ and ε = ε₀)

Thus equation of υ_p will become

υ_p = w/ √ w₂μ₀ε₀ – w₂μ₀ε₀ = w/ √ w√μ₀ε₀ √1-w²_c/w²

υ_p = υ₀ /√1-(f_c/f)²

(υ₀ =1/ √μ₀ε₀ , w= 2πf and w_c = 2πf_c)

Or                                    υ_p = υ₀ / √1- ג²/ ג²_c                      (ג µ 1/f)

These are the expression for the phase velocity.

Phase velocity is also known as Guide Velocity.

These are the characteristics or properties of transverse electric and magnetic waves in parallel planes

The post More characteristics or properties of transverse electric and magnetic waves in parallel planes first appeared on Winner Science.

(a) Propagation Constant in parallel planes

γ_g  = √ K²_g – w²με

γ_g  = √(mπ /α)² – w²με

Where                      K_g = mπ /α

At very high frequency, so that

w²με >> (mπ /α)²

Thus   γ_g  = √-[-( mπ /α)² + w²με ]

γ_g  = √-[ w²με – ( mπ /α)² ]

This shows that quantity under the radical will be negative and then γ_g  will be pure imaginary that is

γ_g  = j √ w²με – (mπ /α)²

Also               γ_g  = α_g + jb_g

Where α_g is attenuation constant and b_g is phase constant.

Definition of  attenuation constant α_g  in parallel planes: α_g is defined as a constant which indicates the rate at which the wave amplitude reduces as it propagates from one point to another.

It is real part of propagation constant. It has units of  dB/m or Neper/m.

Definition of  phase shift constant  in parallel planes b_g . b_g is defined as a measure of the phase shift in radians per unit length.

It is imaginary part of propagation  constant, g_g  with units      radians/m.

Comparing the imaginary parts of above two equations, we get

Phase Constant

b_g = √w²με – ( mπ /α)² ]

Under  these conditions, the fields will progress in the +z direction as waves and the attenuation of such waves will be zero for perfectly conducting planes

that is Attenuation Constant α_g = 0

(b) Cut-Off Frequency

As the frequency is decreased, there will be a stage at critical frequency,

f_c = w_c/2π, at which

w²με = ( mπ /α)²

or                                     w_c = ( mπ /α) 1/√ με

or                                     2πf_c = ( mπ /α) 1/√με

or                                     f_c = (m/2 α) (1/√ με) = (m/2 α) ( 1/√ μ₀ε₀)  (if μ = μ₀, ε = ε₀)

or             f_c = m/2 α υ₀

Here f_c is the cut – off frequency.

For all frequencies less than f_c , the quantity under the radical of equation will be positive and γ_g  will be a real number, that is γ_g  = α_g + j 0 = α_g, as b_g = 0. This implies that fields will be attenuated exponentially in the +z direction and there will be no wave motions as b_g = 0.

Definition of Cut-off Frequency (f_c ). The frequency at which wave motion cases is known as the cut-off frequency of the guide.

Another Definition. It is defined as a frequency below which there exists only attenuation constant, α_g and phase shift constant, b_g = 0 and above which α_g = 0 and γ_g exists.

As                        f_c = m/2 α υ₀

Thus for each value of m, there is a corresponding cut-off frequency below which wave propagation cannot occur. Above the f_c , the wave propagation does occur and there will be no attenuation (α_g = 0) of the wave for perfectly conducting planes.

I will discuss more characteristics or properties of TE and TM waves in parallel planes in next article.

The post CHARACTERISTICS OF TRANSVERSE ELECTRIC AND TRANSVERSE ELECTRIC WAVES first appeared on Winner Science.

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,

H_z = 0, E_z not equals to 0

By substituting Hz = 0 in equation, we get

H_x = 0, E_y = 0,and E_x not equals to 0, H_y not equals to 0

Now write wave equation for free space in term of H

ѲH =d

ѲH =w²μεH                           [.  . =g²_g(jwμ) (σ +jwε)

As σ = 0, then g²g=-w²με]

For the y-component, wave equation will become

d²H_g/dx² + d²H_g/dy² + d²H_g/dz² = – w²μεH_y

As      dH_y/dy = 0 and d²H_g/dz² =  g²_gH_y               [using assumptions (e) and (f) of article “waves between parallel planes“]

The wave equation becomes

d²H_g/dx² + g²_gH_y = – w²μεH_y

or                   d²H_g/dx² = -(g²_g + w²με) H_y

or                   d²H_g/dx² = – K²_g H_y                                                                ..(14) (The equation number is continued from article of “transverse electric waves“)

where                       K²_g = Y²_g + w²με

As                  H_y = H_y0  e^– g_g^z                                                                                    ..(15(a))

d²H_g/dx² =( d²H_g0/dx² ) e^– g_g^z                                                           ..(15(b))

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

d²H_g0/dx² = – K²_g H_y0

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

H_g0 = A₃ sin K_gx + A₄ cos K_gx

                                   H_g =( A₃ sin K_gx + A₄ cos K_gx) e^– g_g^z                         ..(16)

                                                                                               (because H_g = H_g0 e^– g_g^z)

where A_{3 and} A₄ are arbitrary constants.

Here, the boundary conditions cannot be applied directly to H_g to determine the constants  A_{3 and} A₄ because, the tangential component of H is not zero at the surface of a conductor (H_tan ¹ 0). However from equation E_z will be obtained in terms of H_g , and then the boundary conditions would be applied to  E_z

Differentiate equation w.r.t.  x

dH_g/dx = K_g (A₃ cos K_gx – A₄ sin K_gx) e^– g_g^z

From equation

E_z = (1/ jwε) dH_y/d_x

Put value of dH_g/dx in above equation

E_z =( K_g/ jwε) [A₃ cos K_gx – A₄ sin K_gx) e^– g_g^z                                 ..(17)

Applying the boundary condition

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

0 =( K_g/ jwε) [A₃ cos K_g0 – A₄ sin K_gx0) e^– g_g^z

Equation reduces to

E_z =( K_g/ jwε) [ – A₄ sin K_gx) e^– g_g^z                                                    ..(18)

Now applying boundary condition

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

0 =( K_g/ jwε) [- A₄ sin K_gα) e^– g_g^z

Or                              sin K_gα = 0

Or                              K_ga = mπ

Or                              K_g = mπ/α

Equation becomes

E_z =( -K_g A₄/ jwε) [ sin (mπ x/α)] e^– g_g^z

E_z = – mπ x / jwεα   A₄ sin (mπ x/α)] e^– g_g^z                                                  (g_g = jb_g)          ..(19(a))

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

(dH_y/d_g)d_x = jwε (=K_gA₄/ jwε) ( [ sin (mπ x/α)] e^– g_g^z)dx

Or                  H_y = A₄ (mπ /α) (a/mπ ) cos (mπx /α) e^– g_g^z

Or                  H_y = A₄ cos (mπx /α) e^-jb_g^z                                                               ..(19(b))

From              E_x = g_gH_y

Put equation in above equation

                       E_x = (g_g/ jwε) A₄ cos (mπ x/α) e^– g_g^z

                       E_x = (b_g/ wε) A₄ cos (mπ x/α) e ^-jb_g^z                                                           ..(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. E_x and H_y. 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.

The post Transverse magnetic waves in parallel planes first appeared on Winner Science.

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

E_z = 0 and H_z is not equal to 0

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

E_x= 0 and H_y = 0 and

E_{y not equals to} 0 , H_x  not equals to 0

Now write wave equations for free space in terms of E

ѲE =g²_gE

= -w²μεE                     (because g²_g = (jwμ) (σ + jwε) As σ =0(from                                                                            assumption  (c) of article “waves between parallel planes” => g²_g =-w²με)

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

For the y component, the wave equation will become

d²E_y/dx² + d²E_y/dg ² + d²E_y/dz² = -w²μεE_y

also                       dE_g/d_g  = 0                                                      [using assumption (e)]

and                       E_g = E_g0 e^– g_g^z                                                                    [using assumption (f)] of article “waves between parallel planes”

Thus           d²E_g/dz² = g²Eg²_gEg²_gE_g

By substituting above values of dE_g/d_g and d²E_y/dz² in wave equation, we get

d²E_g/dx²  + g²_gE_g= -w²μεE_g

or              d²E_g/dx²  = -(g²_g + w²με) E_g

or              d²E_g/dx²  = –K²_g E_g                                                                  ..(9)

where      K²_g = g²_g + w²με

as              E_g = E_g0  e^– g_g^z                                                                          ..(10(a))

thus           d²E_g/dx²  = d²E_g0 e^– g_g^z/dx²                                                                    ..(10(b))

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

d²E_g0/ dx² = K²_g E_g0 

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

E_g0 = A₁ sin K_gx + A₂cos K_gx

Or              E_g = (A₁ sin K_gx + A₂ cos K_gx) e^– g_g^z                                      ..(11)

Where A₁ and A₂ are arbitrary constants.

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

E_tan = 0 at the surface of conductor

This implies that

                              E_y = 0 at x = 0

E_y = 0 at x = a

Thus apply boundary condition

That           E_y = 0 at x = 0

So equation (11)will become

0 = (A₁ sin K_g0 + A₂ cos K_g0) e^– g_g^z

Thus equation (11) reduces to

E_y = A₁ sin K_gxe^– g_g^z                                                   ..(12)

Now apply boundary condition

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

0 = A₁ sin K_gae^– g_g^z

Or              sinK_ga = 0 as A₁¹ 0

Or              K_ga = mπ

Or              K_g    = 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

. .         E_y = A₁ sin (mπ/a)e^– g_g^z

E_y = A₁ sin (mπ/a)e^–jb^gz       (. . g_g =  jb_g  if α_g = 0)             ..(13(a))

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

A₁ sin (mπx/a)e^– g_g^z = jwμ/ K²_g dH_z/dx   dx

H_z = A₁ K²_g e^– g_g^z /jwμ          sin(mπx/a)/dx

= – A1m²π²/a2   e^– g_g^z/ jwμ  cos (mπx/a)/ mπ      where K²_g = m²π²/a²

H_z =- A₁ mπ cos (mπx/a) e^– g^gz/ jwμ

or                H_z =- A₁ mπ cos (mπx/a) e^– g^gz/ jwμ                       (g_g = jb_g)          ..(13(b))

Differentiate (13(b))w.r.t.  x

dH_z/dx  = A1m²π² sin (mπx/a) e^– g^gz/ jwμa²

Substitute above equation in equation

H_x=- g_g / K²_g A1 m²π²/ jwμa² sin (mπx/a) e^– g_g^z

H_x=- g_g m²π²/a²         A1 m2π2/jwμa2     sin (mπx/a) e^– g_g^z

H_x=- g_g  A₁ /jwμ   sin (mπx/a) e^– g_g^z

H_x=- b_g  A₁ /wμ   sin (mπx/a) e^– g_g^z                 ( g_g =  jb_g)        ..(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 TE_m 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.

The post Transverse electric waves first appeared on Winner Science.

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_gis  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 + ^jb_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 = jwεE   (Modified Ampere’s circuital law)           (1)

                        Ñ x E = jwμH  (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    = jwε (E_xa_x + E_ya_y + E_za_z)

            H_X           H_Y           H_Z

            a_x    dH_z/dg – dH_y/dz    – a_y    dH_z/dx– dH_x/dz    + a_y    dH_y/dx– dH_x/dg

            = jwεE_xa_x +jwεE_ya_y + jwεE_za_z

Comparing the respective components on both sides, we get

            dH_z/dg  –  dH_y/dz  = jwεE_x

            dH_x/dz  –  dH_z/dx  = jwεE_y

            dH_y/dx  –  dH_x/dg = jwεE_z                                                                      (3)

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

            dE_z/dg –  dE_y/dz  = jwμH_x

            dE_x/dz  –  dE_z/dx  = jwμH_y

            dE_y/dx  –  dE_x/dg  = jwμH_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     dH_x /dz =^-y_gH_x0 e^-y_g^z

            Or        dH_x /dz = -g_gH_x                                                                             (5a)

Similarly          dH_y/dz = -g_gH_y                                                                               (5b)

                        dE_x/dz = -g_gE_x                                                                                                (5c)

and                 dE_y/dz = -g_gE_y                                                                                                (5d)

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

            dH_z/dg –  dH_x/dg = dE_z/dg = dE_x/dg = 0                              (5e)

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

            g_gH_y = jwεE_x                                                                                                    (6a)

            -g_gH_x – dH_z/dx = jwεE_y                                                                              (6b)

            dH_y/dx = jwεE_z                                                                                             (6c)                                      

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

            g_gE_y = – jwμH_x                                                                                                                (7a)

_–g_gE_x-dE_z/dx = -jwμH_y                                                                                                               (7b)

dE_y/dx = – jwμH_z                                                                                                         (7c)

Now use equations 6a and 7b

From equation 6a

Ex = g_gH_y/jwε

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

g²_gH_y/jwε + dEz/dx = jwμH_y

            dE_z/dx =( jwμ – g²g/jwε)H_y

jwε (dE_z/dx) = (-w²με – g²g) H_y

 jwε (dE_z/dx) = -H_y(g²g+ H_y w²με)

            = – H_yK²g

            K²g = g²g + w²με

            H_y = – jwε/ K²g   dE_z/dx                                                        (8a)

Again use equations 6a and 7b

From equation

                        H_y = 1/ jwμ ( dE_z/dx  + g_gE_x)

Substituting value of H_y in equation , we have

            g_g/ jwμ ( dE_z/dx  + g²gEx/jwμ )= jwE_x

            g_g/ jwμ ( dE_z/dx ) (jwε- g2g/ jwμ)E_x

            g_g(dE_z/d_x) = (-w² με – g2g) E_x

                _– Y_g(dE_z/d_x) = E_xK²_g

            K²_g = g²g + w² με

            E_x = (gg/ K²_g) dE_z/dx                                                 (8b)

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

            H_x = (-gg/ K²_g) dH_z/dx                                               8c

and     E_y= (jwμ/ K²_g) dH_z/dx                                                           8d

where            K²_g = g²_g + w²με

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 :

1. Transverse Electric (TE) Waves or H Waves (E_z = 0, H_z ¹ 0)
2. Transverse Magnetic (TM) Waves or E Waves (H_z = 0, E_z ¹ 0)

The post Waves between parallel planes first appeared on Winner Science.

Statement. This theorem states that the cross product of electric field vector, E and magnetic field vector, H at any point is a measure of the rate of flow of electromagnetic energy per unit area at that point, that is

                                                P = E x H

Here P → Poynting vector and it is named after its discoverer, J.H. Poynting. The direction of P is perpendicular to E and H and in the direction of vector E x H

Proof. Consider Maxwell’s fourth equation (Modified Ampere’s Circuital Law), that is

                                    del x H = J + ε dE/dt

or                                 J = (del x H)  –  ε dE/dt

The above equation has the dimensions of current density. Now, to convert the dimensions into rate of energy flow per unit volume, take dot product of both sides of above equation by E, that is

                                    E. J = E. (del x H) – εE.  dE/dt         (1)

Use vector Indentity

            del. (E x H) = H. (del x E) – E. (del x H)

or         E. (del x H) = H. (del x E) – del (  E x H )

By substituting value of E. (del x H) in equation (1) , we get

                        E. J. =H . (del x E) – del . (E x H) – εE  dE/dt                     (2)

also from Maxwell’s  third equation (Faraday’s law of electromagnetic induction).

                        del x E =  μdH/dt

By substituting value of del x E in equation (2) we get

                        E. J. =μH . (dH)/dt  – εE.  dE/dt –  del . (E x H)                   (3)

We can write

                        H. dH/dt  = 1/2   dH²/dt                                 (4a)

                        E. dE/dt  = 1/2  dE²/dt                                               (4b)    

By substituting equations 4a and 4b in equation 3 , we get

            E. J. =  -μ/2  dH ²/dt  –   ε/2    dE ²/dt  – del . (E x H)

               E. J. =  -d/dt  [ dH ²/2  +   εE² /2] – del . (E x H)

By taking volume integral on both sides, we get

            ∫E. J. dV =  -d/dt  ∫ [μ H ² /2 +   εE²/2 ]  dV – ∫del . (E x H) dV      (5)

apply Gauss’s Divergence theorem to second term of R.H.S., to change volume integral into surface integral, that is

            ∫del . (E x H) dV = ∫ (E x H) . dS

Substitute above equation in equation 5

Thus  

            ∫E. J. dV = -d/dt  ∫[ ε E²/2 +  μ H²/2]  dV – ∫(E x H) . dS     (6)

or        ∫ (E x H) . dS = ∫-d/dt   [ ε E ² /2 +  μ H ² /2]  dV –∫ E. J. dV

Interpretation of above equation :

L.H.S. Term

            ∫ (E x H) . dS  → It represents the rate of outward flow of energy from a volume S

V and the integral is over the closed surface surrounding the volume. This rate of outward flow of power from a volume V is represented by 

                        ∫ P . dS = ∫ (E x H) . dS

where Poynting vector, P = E x H

Inward flow of power is represented by

                        – ∫ P . dS = – φ∫ (E x H) . ds

R.H.S. First Term

                        -d /dt  [μ H ²/2   +  ε E ² /2]  dV → If the energy is flowing out of the region, there must be  a corresponding decrease of electromagnetic energy. So here negative sign indicates decrease. Electromagnetic energy is the sum of magnetic energy, μ H ²/2 and electric

energy, ε E ²/2 . So first term of R.H.S. represents rate of decrease of stored electromagnetic  2  energy. 

R.H.S. Second Term

∫ (E. J) dV →Total ohmic power dissipated within the volume.

So from the law of conservation of energy, equation (6) can be written in words as

Rate of energy disripation in volume V = Rate at which stored electromagnetic energy is decreasing in V + Inward rate of flow of energy through the surface of the volume.

The post Poynting theorem and derivation first appeared on Winner Science.