Unit - 6

Maxwell’s equations & EMW

6.1 Continuity equation for current densities

Consider a closed surface S enclosing a Volume V through which currents are flowing. Let I be the current passing through the surface at any time t. Consider a small area of the closed surface. The outward current flowing through    at a time t is

Figure 1

DI =       …………. (1)

The total current flowing out through the closed surface S at time t is

I =       …………. (2)

Suppose is ρ the Volume Charge density of charges inside the surface S. Then the total charge inside the Volume V is

q =       …………. (3)

Since the current is flowing outward is it mean that charge within the enclosed surface is decreasing with time. The time rate of decrease of charge is

= -       …………. (4)

The negative sign shows that the charge inside the surface S is decreasing with time. The above equation can be written as

= -       …………. (5)

Since the charge is conserved, the current flowing outward through the closed surface must be equal to the rate of decrease of charge. Thus 

I =

Using equation (2) and (5), it becomes

= -       …………. (6)

According to divergence theorem

= dV

Using this result in equation (6) we obtain

dV = -   

This equation holds good for arbitrary volumes. Therefore the integrands must be equal. Hence  

=0 …………. (7)

This equation is called equation of continuity. It is the mathematical statement of law of conservation of charge. 

SPECIAL CASE

Consider a conductor AB through which a steady current I is flowing. The current I is said to be steady if charge is flowing per unit time through any two any sections C and D is the same. This means that the total amount of charge entering the volume V through C is equal to the charge leaving the volume V through D. Thus there is no change in the volume charge density in the volume V with time i.e.

=0

Figure 2

We may define steady current as that current in which the charge density is independent of time.

Using equation (7) we obtain

This is known as the equation of continuity for steady currents

6.2 Ampere’s law and its modification

Statement of Ampere’s circuital law (without modification). It states that the line integral of the magnetic field H around any closed path or circuit is equal to the current enclosed by the path.

That is                                   ∫ H. DL=I

Let the current is distributed through the surface with a current density J

Then                                                I=

This implies that                           =                          (1)

Apply Stoke’s theorem to L.H.S. Of equation (1) to change line integral to surface integral,

That is                                =

Substituting above equation in equation (9), we get

=

As two surface integrals are equal only if their integrands are equal

Thus,                                            ∇ x H=J                                          (2)

This is the differential form of Ampere’s circuital Law (without modification) for steady currents.

Take divergence of equation (2)

∇.(∇xH)= ∇.J

As divergene of the curl of a vector is always zero, therefore

∇. (∇xH)=0

It means                                     ∇.J=0

Now, this is equation of continuity for steady current but not for time varying fields, as equation of continuity for time varying fields is

∇ .J = –

If we do some simple mathematical tricks to Maxwell's Equations, we can derive some new equations. Here we'll look at the continuity equation, which can be derived from Gauss' Law and Ampere's Law.

We will start with a vector identity which states that the divergence of the curl of any vector field is always zero:

∇. (∇xH) =0        ………… (1)

If we apply the divergence to both sides of Ampere's Law, then we obtain:

∇. ( )= ∇. (∇x H) =0 

= - ∇. J        ………… (2)

If we apply Gauss' Law to rewrite the divergence of the Electric Flux Density (D), we have derived the continuity equation

∇. J = –         ………… (3)

The left side of the equation is the divergence of the Electric Current Density (J). This is a measure of whether current is flowing into a volume (i.e. the divergence of J is positive if more current leaves the volume than enters).

Recall that current is the flow of electric charge. So if the divergence of J is positive, then more charge is exiting than entering the specified volume. If charge is exiting, then the amount of charge within the volume must be decreasing. This is exactly what the right side is a measure of how much electric charge is accumulating or leaving in a volume. Hence, the continuity equation is about continuity - if there is a net electric current is flowing out of a region, and then the charge in that region must be decreasing. If there is more electric current flowing into a given volume than exiting, than the amount of electric charge must be increasing.

6.3 Differential and integral forms of Maxwell’s equation

Maxwell equations are of fundamental importance since they describe the whole of classical electromagnetic phenomena.

From a classical perspective, light can be described as waves of electromagnetic radiation. As such, Maxwell equations are very useful to illustrate a number of the characteristics of light including polarization.

We are just to stating these equations without derivation. Since our goal is simply to apply them, the usual approach will be followed.

Maxwell’s four equations are given by

∇·E = ρ/ε0        (1)

∇×E = −∂B/∂t        (2)

∇×H = J + ∂D/∂t       (3)

∇·B = 0        (4)

These equations illustrate the unique coexistence in nature of the electric field and the magnetic field. The first two equations give the value of the given flux through a closed surface, and the second two equations give the value of a line integral around a loop. In this notation,

∇=(∂/∂x, ∂/∂y, ∂/∂z)

E is the electric vector

B is the magnetic induction

ρ is the electric charge density

j is the electric current density

ε0 is the permittivity of free space

c is the speed of light.

In addition to Maxwell equations, the following identities are useful:

J =  σE         (5)

D = εE         (6)

B = μH         (7)

Here,

D is the electric displacement

H is the magnetic vector

σ is the specific conductivity

ε is the dielectric constant (or permittivity)

μ is the magnetic permeability

In the Gaussian systems of units, Maxwell equations are given in the form of

∇·B=0          (8)

∇·E=4πρ         (9)

∇×H=(1/c)(∂D/∂t+4πj)       (10)

∇×E=−(1/c)(∂B/∂t)        (11)

Free space or non-conducting medium. We know that non conducting medium means no current so conductivity is zero i.e. σ=0

So current density J=σE will also become zero as σ=0 Also free space means no charges which leads to ρ=0. These points mentioned below.

(a)    No condition current i.e σ=0, thus J=0 ( J=σE)

(b)   No charges (i.e ρ=0)

For the case of no charges or currents, that is, j = 0 and ρ = 0, and a homogeneous medium. Using these the Maxwell equation can rewritten as

∇.D=0 or ∇.E=0      as  ρ=0         (12)

∇  x E= -dB/dt  or    ∇ x E= -μ dH/dt    because  B = μH       (13)

∇ x H=d D/dt  or  ∇ x H = ε dE/dt         (J=0) and D = εE  (14)

∇.B=0                                      (15)

Maxwell's Equations are a set of 4 complicated equations that describe the world of electromagnetics. These equations describe how electric and magnetic fields propagate, interact, and how they are influenced by objects.

Maxwell was one of the first to determine the speed of propagation of electromagnetic (EM) waves was the same as the speed of light - and hence to conclude that EM waves and visible light were really the same thing.

These equations are rules the universe uses to govern the behaviour of electric and magnetic fields. A flow of electric current will produce a magnetic field. If the current flow varies with time (as in any wave or periodic signal), the magnetic field will also give rise to an electric field. Maxwell's Equations shows that separated charge (positive and negative) gives rise to an electric field - and if this is varying in time as well will give rise to a propagating electric field, further giving rise to a propagating magnetic field.

From a classical perspective, light can be described as waves of electromagnetic radiation. As such, Maxwell equations are very useful to illustrate a number of the characteristics of light including polarization.

The First Maxwell’s equation (Gauss’s law for electrostatics)

Gauss' Law is the first of Maxwell's Equations which dictates how the Electric Field behaves around electric charges. Gauss' Law can be written in terms of the Electric Flux Density and the Electric Charge Density as:

∇·E = ρ/ε0 

0r

∇·D = ρ 

Hence, Gauss' law is a mathematical statement that the total Electric Flux exiting any volume is equal to the total charge inside. Hence, if the volume in question has no charge within it, the net flow of Electric Flux out of that region is zero. If there is positive charge within a volume, then there exists a positive amount of Electric Flux exiting any volume that surrounds the charge. If there is negative charge within a volume, then there exists a negative amount of Electric Flux exiting (i.e. the Electric Flux enters the volume).

The Gauss’s law states that flux passing through any closed surface is equal to 1/ε0 times the total charge enclosed by that surface.

Integral form of Maxwell’s 1st equation

It is the integral form of Maxwell’s 1st equation.

Maxwell’s first equation in differential form

It is called the differential form of Maxwell’s 1st equation.

The Second Maxwell’s equation (Gauss’s law for magnetism)

The Gauss’s law for magnetism states that net flux of the magnetic field through a closed surface is zero because monopoles of a magnet do not exist.

We know that Gauss' Law for Electric Fields states that the divergence of the Electric Flux Density D is equal to the volume electric charge density. But the second equation, Gauss' Magnetism law states that the divergence of the Magnetic Flux Density (B) is zero.

Why? Why isn't the divergence of B equal to the magnetic charge density?

Well - it is. But it just so happens that no one has ever found magnetic charge - not in a laboratory or on the street or on the subway. And therefore, until this hypothetical magnetic charge is found, we set the right side of Gauss' Law for Magnetic Fields to zero:

∇·B = 0 

Now, you may have played with magnets when you were little, and these magnetic objects attracted other magnets similar to how electric charges repel or attract like electric charges. However, there is something special about these magnets - they always have a positive and negative end. This means every magnetic object is a magnetic dipole, with a north and South Pole. No matter how many times you break the magnetic in half, it will just form more magnetic dipoles. Gauss' Law for Magnetism states that magnetic monopoles do not exist - or at least we haven't found them yet.

Maxwell’s Third equation (Faraday's Law)

Faraday figured out that a changing Magnetic Flux within a circuit (or closed loop of wire) produced an induced EMF, or voltage within the circuit. Maxwell gives his second equation from this.

Faraday's Law tells us that a magnetic field that is changing in time will give rise to a circulating E-field. This means we have two ways of generating E-fields - from Electric Charges (or flowing electric charge, current) or from a magnetic field that is changing.

∇×E = −∂B/∂t        

According to Faraday’s law of electromagnetic induction

It is the differential form of Maxwell’s third equation.

Maxwell’s fourth equation (Ampere's Law)

Ampere's Law tells us that a flowing electric current gives rise to a magnetic field that circles the wire. In addition to this, it also says that an Electric Field that is changing in time gives rise to a magnetic field that encircles the E-field - this is the Displacement Current term that Maxwell himself introduced.

This means there are 2 ways to generate a solenoidal (circulating) H-field - a flowing electric current or a changing Electric Field. Both give rise to the same phenomenon.

The modified form of ampere Law is given by

H.dL=(J+dD/dt)

Take integration on both sides we have

∫H.dL=∫(J+dD/dt).dS

Apply stoke’s theorem to L.H.S. Of above equation, we get

∫( ∇xH).dS=∫ H.dL

Now the above equation is written as

∫( ∇xH).dS =∫(J+dD/dt).dS

By cancelling the surface integral on both sides we have

∇xH =J+dD/dt

Which  is the differential form of Maxwell’s equation.

This can also be written in the form B

The Ampere-Maxwell Law

Begin with the Ampere-Maxwell law in integral form.

∮CB⋅dl=μ0∫SJ⋅dS+μ0ϵ0ddt∫SE⋅dS{\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\mu _{0}\epsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} }Invoke Stokes' theorem.

∫S(∇×B)⋅dS=μ0∫SJ⋅dS+μ0ϵ0ddt∫SE⋅dS{\displaystyle \int _{S}(\nabla \times \mathbf {B} )\cdot \mathrm {d} \mathbf {S} =\mu _{0}\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\mu _{0}\epsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} }Set the equation to 0. ∫S(∇×B)⋅dS−μ0∫SJ⋅dS−μ0ϵ0ddt∫SE⋅dS=0∫S(∇×B−μ0J−μ0ϵ0∂E∂t)⋅dS=0{\displaystyle {\begin{aligned}\int _{S}(\nabla \times \mathbf {B} )\cdot \mathrm {d} \mathbf {S} -\mu _{0}\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} -\mu _{0}\epsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} &=0\\\int _{S}\left(\nabla \times \mathbf {B} -\mu _{0}\mathbf {J} -\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot \mathrm {d} \mathbf {S} &=0\end{aligned}}}

Physical Significance Of Each Equation

Physical significance of Maxwell’s Ist equation

∇·E = ρ/ε0

According to this total electric flux through any closed surface is 1/0 times the total charge enclosed by the closed surfaces, representing Gauss's law of electrostatics, As this does not depend on time, it is a steady state equation. Here for positive , divergence of electric field is positive and for negative  divergence is negative. It indicates that  is scalar quantity.

Physical significance of Maxwell’s 2nd equation

∇·B = 0

It represents Gauss law of magnetostatic as ∇·B = 0 resulting that isolated magnetic poles or magnetic monopole cannot exist as they appear only in pairs and there is no source or sink for magnetic lines of forces. It is also independent of time i.e. steady state equation.

Physical significance of Maxwell’s 3rd equation

∇×E = −∂B/∂t 

It shows that with time varying magnetic flux, electric field is produced in accordance with Faraday is law of electromagnetic induction. This is a time dependent equation.

Physical significance of Maxwell’s 4th equation

∇×H = J + ∂D/∂t 

This is a time dependent equation which represents the modified differential form of Ampere's circuital law according to which magnetic field is produced due to combined effect of conduction current density and displacement current density.

Key Takeaways

• The First Maxwell’s equation is known as Gauss’s law for electrostatics. It is given by ∇·E = ρ/ε0 . According to this total electric flux through any closed surface is 1/0 times the total charge enclosed by the closed surfaces, representing Gauss's law of electrostatics,
• The Second Maxwell’s equation is known as Gauss’s law for magnetism. It is given by ∇·B = 0. The Gauss’s law for magnetism states that net flux of the magnetic field through a closed surface is zero because monopoles of a magnet do not exist.
• Maxwell’s Third equation is known as Faraday's Law. It is given by ∇×E = −∂B/∂t. Faraday figured out that a changing Magnetic Flux within a circuit (or closed loop of wire) produced an induced EMF, or voltage within the circuit.
• Maxwell’s fourth equation is known as Ampere's Law. It is given by ∇×H = J + ∂D/∂t. Ampere's Law tells us that a flowing electric current gives rise to a magnetic field that circles the wire. In addition to this, it also says that an Electric Field that is changing in time gives rise to a magnetic field that encircles the E-field - this is the Displacement Current.

6.4 Maxwell’s equation in vacuum and non-conducting medium

Maxwell Equations

Maxwell equations are of fundamental importance since they describe the whole of classical electromagnetic phenomena.

From a classical perspective, light can be described as waves of electromagnetic radiation. As such, Maxwell equations are very useful to illustrate a number of the characteristics of light including polarization.

We are just to stating these equations without derivation. Since our goal is simply to apply them, the usual approach will be followed.

Maxwell’s four equations are given by

∇·E = ρ/ε0        (1)

∇×E = −∂B/∂t        (2)

∇×H = J + ∂D/∂t       (3)

∇·B = 0        (4)

These equations illustrate the unique coexistence in nature of the electric field and the magnetic field. The first two equations give the value of the given flux through a closed surface, and the second two equations give the value of a line integral around a loop. In this notation,

∇= (∂/∂x, ∂/∂y, ∂/∂z)

E is the electric vector

B is the magnetic induction

ρ is the electric charge density

j is the electric current density

ε0 is the permittivity of free space

c is the speed of light.

In addition to Maxwell equations, the following identities are useful:

J =  σE         (5)

D = εE         (6)

B = μH         (7)

Here,

D is the electric displacement

H is the magnetic vector

σ is the specific conductivity

ε is the dielectric constant (or permittivity)

μ is the magnetic permeability

In the Gaussian systems of units, Maxwell equations are given in the form of

∇·B=0          (8)

∇·E=4πρ         (9)

∇×H=(1/c) (∂D/∂t+4πj)       (10)

∇×E=−(1/c) (∂B/∂t)        (11)

Free space or non-conducting medium. We know that non conducting medium means no current so conductivity is zero i.e. σ=0

So current density J=σE will also become zero as σ=0 Also free space means no charges which leads to ρ=0. These points mentioned below.

(a)    No condition current i.e. σ=0, thus J=0 (J=σE)

(b)   No charges (i.e. ρ=0)

For the case of no charges or currents, that is, j = 0 and ρ = 0, and a homogeneous medium. Using these the Maxwell equation can rewritten as

∇. D=0 or ∇. E=0      as ρ=0        (12)

∇ x E= -dB/dt  or    ∇ x E= -μ dH/dt    because B = μH       (13)

∇ x H=d D/dt  or  ∇ x H = ε dE/dt    (J=0) and D = εE  (14)

∇.B=0                                      (15)

Now taking curl of second Maxwell’s equation (13), we get

∇ x (∇ x E) =- μ d/dt (∇ x H)

Applying standard vector identity, that is [∇ *(∇*E) =∇ (∇. E)-∇2E] on left hand side of above equation, we get

∇ (∇ .E)-∇2E= -μ d/dt (∇ x H)                                                           (16)

Substituting equations (13) and (14) in equations (16), we get

-∇2E= – με d/dt (dE/dt)

Or                               ∇2E=με d 2 E/dT2       (17)

Equation (17) is the required wave equation in terms of electric field intensity, E for free space. This is the law that E must obey.

WAVE EQUATION IN TERMS OF MAGNETIC FIELD INTENSITY, H

Take curl of fourth Maxwell’s equation (14), we get

∇x(∇xH) =ε d/dt(∇xE)

Applying standard vector identity that is

[∇*(∇*H) =∇ (∇.H)-∇2H]

On left side of above equation, we get

∇(∇.H)-∇2H= ε d/dt(∇xE)                                      (18)

Substituting equations (14) and (13) in equation (18), we get

-∇2H= – μεd/dt(dH/dt)

Or                      

∇2H=με d2H/dt2        (19)

Equations (19) is the required wave equation in terms of magnetic field intensity, H and this is the law that H must obey

For vacuum μ=μ0 and ε=ε0, equations (17) and (19) will become

∇2 E=μ0ε0 d2E/dt2        (20)

And     ∇2H=    μ0ε0 d2H/dt2        (21)

This leads to an expression for the velocity of propagation

From equation both equations (20) and (21) have the form of the general wave equation for a wave

(x, t) traveling in the x direction with speed v. Equating the speed with the coefficients, we derive the speed of electric and magnetic waves, which is a constant that we symbolize with “c”

It is useful to note that in vacuum

c2=1/ε0μ0        

Where μ0 is the permeability of free space

6.5 The wave equation

Wave is nothing but a pattern of disturbance which propagates and carry energy with it. You can produce a wave on a rope by moving one end of the rope up and down. The wave produces on rope needs a medium to propagate and here medium is rope itself. This type of waves is known as mechanical waves. But in the case of Electromagnetic waves, they don't need a medium to propagate. Electromagnetic waves are waves that are created as a result of variations of electric field and a magnetic field. Or we can say that Electromagnetic waves are nothing but changing magnetic and electric fields. Electromagnetic waves are also known to be solutions of Maxwell's equations. And Maxwell's equations are the fundamental equations of electrodynamics. Electromagnetic waves can transmit energy and travel through a vacuum.

Light waves are examples of electromagnetic waves. Generally, Electromagnetic waves are shown by a sinusoidal graph.

Figure 3

As shown in figure Electromagnetic waves consist of time-varying electric and magnetic fields and they are perpendicular to each other and these both fields are also perpendicular to the direction of propagation of waves.

Maxwell’s four equations are given by

∇·E = ρ/ε0        (1)

∇×E = −∂B/∂t        (2)

∇×H = J + ∂D/∂t       (3)

∇·B = 0        (4)

These equations illustrate the unique coexistence in nature of the electric field and the magnetic field. The first two equations give the value of the given flux through a closed surface, and the second two equations give the value of a line integral around a loop. In this notation,

∇= (∂/∂x, ∂/∂y, ∂/∂z)

E is the electric vector

B is the magnetic induction

ρ is the electric charge density

j is the electric current density

ε0 is the permittivity of free space

c is the speed of light.

In addition to Maxwell equations, the following identities are useful:

J = σE         (5)

D = εE         (6)

B = μH         (7)

Here,

D is the electric displacement

H is the magnetic vector

σ is the specific conductivity

ε is the dielectric constant (or permittivity)

μ is the magnetic permeability

Free space or non-conducting medium. We know that non-conducting medium means no current so conductivity is zero i.e. σ=0

So current density J=σE will also become zero as σ=0 Also free space means no charges which leads to ρ=0. These points mentioned below.

(a)    No condition current i.e σ=0, thus J=0 (J=σE)

(b)   No charges (i.e. ρ=0)

For the case of no charges or currents, that is, j = 0 and ρ = 0, and a homogeneous medium. Using these the Maxwell equation can rewritten as

∇. D=0 or ∇. E=0      as ρ=0        (12)

∇ x E= -dB/dt  or    ∇ x E= -μ dH/dt    because B = μH       (13)

∇ x H=d D/dt  or  ∇ x H = ε dE/dt        (J=0) and D = εE  (14)

∇.B=0                                      (15)

WAVE EQUATION IN TERMS OF ELECTRIC FIELD INTENSITY, E

Now taking curl of second Maxwell’s equation (13), we get

∇ x (∇ x E) =- μ d/dt (∇ x H)

Applying standard vector identity, that is [∇ *(∇*E) =∇ (∇. E)-∇2E] on left hand side of above equation, we get

∇ (∇ .E)-∇2E= -μ d/dt (∇ x H)                                                           (16)

Substituting equations (13) and (14) in equations (16), we get

-∇2E= – με d/dt (dE/dt)

Or                               ∇2E=με d 2 E/dT2       (17)

Equation (17) is the required wave equation in terms of electric field intensity, E for free space. This is the law that E must obey.

WAVE EQUATION IN TERMS OF MAGNETIC FIELD INTENSITY, H

Take curl of fourth Maxwell’s equation (14), we get

∇x(∇xH) =ε d/dt(∇xE)

Applying standard vector identity that is

[∇*(∇*H) =∇ (∇.H)-∇2H]

On left side of above equation, we get

∇(∇.H)-∇2H= ε d/dt(∇xE)                                          (18)

Substituting equations (14) and (13) in equation (18), we get

-∇2H= – μεd/dt(dH/dt)

Or                      

∇2H=με d2H/dt2        (19)

Equations (19) is the required wave equation in terms of magnetic field intensity, H and this is the law that H must obey

For vacuum μ=μ0 and ε=ε0, equations (17) and (19) will become

∇2 E=μ0ε0 d2E/dt2        (20)

And     ∇2H=    μ0ε0 d2H/dt2        (21)

This leads to an expression for the velocity of propagation

From equation both equations (20) and (21) have the form of the general wave equation for a wave

(x, t) traveling in the x direction with speed v. Equating the speed with the coefficients, we derive the speed of electric and magnetic waves, which is a constant that we symbolize with “c”

It is useful to note that in vacuum

c2=1/ε0μ0        

Where μ0 is the permeability of free space

Let us rewrite the equation (20) and (21) for one dimension.

Or                              d 2 E/dx2 = μ0ε0 d 2 E/dt2   (22)

d2H/dx2 = μ0ε0 d2H/dt2   (23)

The simplest solutions to the differential equations (22) and (23) are sinusoidal wave functions:

E(x) = Emax cos(kx-t)      (24)

B(x) = Bmax cos(kx-t)     (25)

Where k = 2π/λ is the wavenumber, ω = 2πƒ is the angular frequency, λ is the wavelength, f is the frequency and ω /k=f=v= c.

6.6 Plane electromagnetic waves in vacuum, transverse character

PLANE ELECTROMAGNETIC WAVES IN VACUUM

Plane Monochromatic Waves in Non-conducting Media One of the most important consequences of the Maxwell equations is the equations for electromagnetic wave propagation in a linear medium. In the absence of free charge and current densities the Maxwell equations are

    ………. (1)

The wave equations for E and B are derived by taking the curl of  and an

     ………. (2)

For uniform isotropic linear media, we have D = and B=μH, where and are = and μ in general complex functions of frequency ω. Then we obtain

     ………. (3)

Since

   ………. (4)

And similarly

      ………. (5)

Using these identities we have

      ………. (6)

Monochromatic waves may be described as waves that are characterized by a single frequency. Assuming the fields with harmonic time dependence e-iωt, so that E (x, t) =E(x) e-iωt and B (x, t) =B(x) e-iωt we get the Helmholtz wave equations

      ………. (7)

For vacuum, so 0 and μ= μ0. Further, suppose E(x) varies in only one dimension, say the z-direction, and is independent of  x and y . Then Equation (7) becomes

     .……. (8)

Where the wave number k=ω/c. This equation is mathematically the same as the harmonic oscillator equation and has solutions

      .……. (9)

Where is a constant vector? Therefore, the full solution is

   .……. (10)

This represents a sinusoidal wave traveling to the right or left in the z-direction with the speed of light c. Using the Fourier superposition theorem, we can construct a general solution of the form

   .……. (11)

TRANSVERSE NATURE

In an electromagnetic wave, electric and magnetic field vectors are perpendicular to each other and at the same time are perpendicular to the direction of propagation of the wave. This nature of electromagnetic wave is known as Transverse nature.

Maxwell proved that both the electric and magnetic fields are perpendicular to each other in the direction of wave propagation. He considered an electromagnetic wave propagating along positive x-axis. When a rectangular parallelepiped was placed parallel to the three co-ordinate axis, the electric and magnetic fields propagate sinusoidal with the x-axis and are independent of y and z-axis.

A wave in which the values of variable are constant in a plane perpendicular to the direction of propagation of the wave is called plane wave. These planes may also term as wave fronts.

In this section we will study the variation of field of and of an em plane wave with space and time.

Variation with Space

First we will consider the variation with space. Suppose a plane em wave is propagating along x axis. Then the values of the field vectors and     will be constant on any plane parallel to YZ plane. That is: 

For

= = = = 0       ………… (1)

For

= = = = 0       ………… (2)

From Maxwell’s equation for free space, we have

∇.E=0      

Therefore

++ =0

Which gives

=0          ………… (3)

Also, From Maxwell’s equation for free space, we have

∇. B=0   we have ∇.H=0   as B = μH 

Hence

++ =0

Which gives

=0         ………… (4)

The equation (3) and (4) show that there is no variation of and along the X axis. In other words, there is no variation in the longitudinal component of and .

Time Variation of and

The Maxwell’s equation in free space can be written as:

∇ x E= -dB/dt  or    ∇ x E= -μ  because  B = μH  

Comparing the rectangular components, we find:

- = -μ         ………… (5)

- = -μ         ………… (6)

- = -μ         ………… (7)

Applying the condition listed in equation (1), from equation (5), we find:

- =0

=0          ………… (8)

Again using Maxwell’s equation for free space is:

∇ x H=   = Ɛ

Comparing the rectangular components, we can write:

- = Ɛ         ………… (9)

- = Ɛ        ………… (10)

- = Ɛ        ………… (11)

Again using the conditions listed in equation (2) gives:

- = 0

= 0          ………… (12)

From equation (8) and (12), we find that there is no variation in the values of and with time. That is and neither vary with space nor with time. So at the most they can have constant value. But the constant values of and contribute nothing towards the wave because for the wave the field vectors must possess oscillatory nature. This shows that there is no longitudinal component of the field vectors in the electromagnetic wave. In other words, for the purpose of discussing of the wave nature, we may put  =0

Now from equation 6 and 7 by putting = 0, we find:

= μ          ………… (13)

= - μ          ………… (14)

Similarly putting =0 in equation (10) and (11) we find:

= - Ɛ        ………… (10)

= Ɛ        ………… (11)

These relations show that in the em values and are related to each other. Also and are related to each other, and their time or space variation are not zero. From this we conclude that the em wave is transverse in nature.

6.7 Relation between electric and magnetic fields of an electromagnetic wave

We know that wave equations are given by 

∇2 E=με d2E/dt2         …………. (1)

And     ∇2H=    με d2H/dt2         …………. (2)

The electric and magnetic vectors in an em waves are given by:

= E0eik (vt-x)         …………. (3)

= H0eik (vt-x)         …………. (4)

It is evident therefore that:

• Both electric and magnetic vectors oscillate sinusoidally with the same frequency and in the same phase with each other.
• The planes of their oscillations are mutually perpendicular and also perpendicular to the direction of propagation of the waves (transverse wave).
• The disturbance corresponding to both the electric and magnetic vectors propagate with the same speed and the waves corresponding to each have the same length.
• The relation between their time and space variations is given by:

= - μ         …………. (5)

And

= - Ɛ        …………. (6)

Substitute the values of and in equation (5), we find:

-ik E0eik (vt-x) =- μ (ikv) H0eik (vt-x) 

Which gives

E0 =- μvH0

But v =

E0 = μ x H0

E0 = H0

= = =

That is the relation between the values of the electric and magnetic vectors as determined by the relative values μ and Ɛ. In other words, the ratio of the electric and magnetic vectors is directly proportional to the square root of the ratio of μ and Ɛ.

For free space or vacuum

= = = = 376.73 377

E0 =377 H0 or E =377H

It shows that the values of electric vector at any instant in the em wave are about 377 times the values of magnetic vector. It is because of this reason that while discussing the behaviour of light as em wave, we prefer the use of the electric vector.

6.8 Energy in an electromagnetic field

Here we will describe how much energy there is in any volume element of space, and also the rate of energy flow. Suppose we think first only of the electromagnetic field energy. We will let u represent the energy density in the field (that is, the amount of energy per unit volume in space) and let the vector S represent the energy flux of the field (that is, the flow of energy per unit time across a unit area perpendicular to the flow). Then, in perfect analogy with the conservation of charge, we can write the “local” law of energy conservation in the field as

= −r· S      …………… (1)

Of course, this law is not true in general; it is not true that the field energy is conserved. Suppose you are in a dark room and then turn on the light switch. All of a sudden the room is full of light, so there is energy in the field, although there wasn’t any energy there before. Equation (1) is not the complete conservation law, because the field energy alone is not conserved; only the total energy in the world—there is also the energy of matter. The field energy will change if there is some work being done by matter on the field or by the field on matter.

However, if there is matter inside the volume of interest, we know how much energy it has: Each particle has the energy m0c2/

The total energy of the matter is just the sum of all the particle energies, and the flow of this energy through a surface is just the sum of the energy carried by each particle that crosses the surface.

We want now to talk only about the energy of the electromagnetic field. So we must write an equation which says that the total field energy in a given volume decreases either because field energy flows out of the volume or because the field loses energy to matter (or gains energy, which is just a negative loss). The field energy inside a volume V is

V         …………… (2)

And its rate of decrease is minus the time derivative of this integral. The flow of field energy out of the volume V is the integral of the normal component of S over the surface ∑ that encloses V,

∑        …………… (3)

So

- V = ∑ + work done on matter inside V  …………… (4)

We have seen before that the field does work on each unit volume of matter at the rate E · j. [The force on a particle is F = q (E + v × B), and the rate of doing work is F · v = qE · v. If there are N particles per unit volume, the rate of doing work per unit volume is NqE · v, but Nqv = j.] So the quantity E · j must be equal to the loss of energy per unit time and per unit volume by the field. Equation (27.3) then becomes

- V = ∑ + V     …………… (5)

This is our conservation law for energy in the field. We can convert it into a differential equation like Eq. (1) if we can change the second term to a volume integral. That is easy to do with Gauss’ theorem. The surface integral of the normal component of S is the integral of its divergence over the volume inside. So Eq. (4) is equivalent to

-         V = V  + V     ……………(6)

Where we have put the time derivative of the first term inside the integral. Since this equation is true for any volume, we can take away the integrals and we have the energy equation for the electromagnetic fields:

- =         …………… (7)

Now this equation doesn’t do us a bit of good unless we know what u and S are. Perhaps we should just tell you what they are in terms of E and B, because all we really want is the result. However, we would rather show you the kind of argument that was used by Poynting in 1884 to obtain formulas for S and u, so you can see in next article.

6.9 Poynting theorem

When electromagnetic wave travels in space, it carries energy and energy density is always associated with electric fields and magnetic fields.

The rate of energy travelled through per unit area i.e. the amount of energy flowing through per unit area in the perpendicular direction to the incident energy per unit time is called Poynting vector.

Mathematically Poynting vector is represented as

= =( )   (i)

The direction of Poynting vector is perpendicular to the plane containing   and . Poynting vector is also called as instantaneous energy flux density. Here rate of energy transfer is perpendicular to both   and.  It represents the rate of energy transfer per unit area.

UNIT

Its unit is W/m2.

Poynting Theorem

Poynting theorem states that the net power flowing out of a given volume V is equal to the time rate of decrease of stored electromagnetic energy in that volume decreased by the conduction losses.

i.e. total power leaving the volume =Rate of decrease of stored electromagnetic energy – ohmic power dissipated due to motion of charge

Proof

The energy density carried by the electromagnetic wave can be calculated using Maxwell's equations

Div = 0      ...(i)

Div =0      ...(ii)

Curl = -      …(iii)

Curl = +     …(iv)

Taking scalar product of (3) with H and (4) with

i.e.  curl = -    ……(v)

And curl = + .    …..(vi)

Doing (vi) – (v) i.e.

So from equation (vii)

Or

Integrating equation (viii) over a volume V enclosed by a Surface S

Total power leaving the volume = rate of decrease of stored e.m. Energy -ohmic power dissipated due to charge motion

This equation (ix) represents the Poynting theorem according to which the net power flowing out of a given volume is equal to the rate of decrease of stored electromagnetic energy in that volume minus the conduction losses.

In equation (ix) .ds represents the amount of electromagnetic energy crossing the closed surface per second or the rate of flow of outward energy through the surface S enclosing volume V i.e. it is Poynting vector.

The term  represent

The energy stored in electric and magnetic fields respectively and their sum denotes the total energy stored in electromagnetic field. So total terms gives the rate of decrease of energy stored in volume V due to electric and magnetic fields.

Gives the rate of energy transferred into the electromagnetic field.

This is also known as work-energy theorem. This is also called as the energy conservation law in electromagnetism.

Key Takeaways

• The rate of energy travelled through per unit area i.e. the amount of energy flowing through per unit area in the perpendicular direction to the incident energy per unit time is called Poynting vector.
• Mathematically Poynting vector is represented as = =( ) 
• The direction of Poynting vector is perpendicular to the plane containing   and .
• Poynting vector is also called as instantaneous energy flux density. It represents the rate of energy transfer per unit area.
• Its unit is W/m2.

References:

1. Fundamentals of Physics Electricity and Magnetism, Halliday and Resnick, tenth edition (published 2013).

2. Electricity, magnetism and light, W. Saslow, 1st edition

3. Electromagnetic Theory, Singh and Prasad, I. K. International Publication, 1/e

4. Electricity, Magnetism & Electromagnetic Theory, S. Mahajan and Choudhury, 2012, Tata McGraw

5. Elements of Electromagnetics, M.N.O. Sadiku, 2010, Oxford University Press