Unit - 6

Maxwell’s equations & EMW

Q1) Explain Ampere’s Law and its modification.

A1) 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 divergence 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.

Q2) Explain Differential and integral forms of Maxwell’s equation.

A2) 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.

Q3) Explain the First Maxwell’s equation (Gauss’s law for electrostatics).

A3) 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.

Q4) Explain the Second Maxwell’s equation (Gauss’s law for magnetism).

A4) 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.

Q5) Explain Maxwell’s Third equation (Faraday's Law).

A5) 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.

Q6) Explain Maxwell’s fourth equation (Ampere's Law).

A6) 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

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}}}

Q7) Explain Maxwell’s equation in vacuum and non-conducting medium.

A7) 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.

Q8) Explain Wave equation in terms of Magnetic field intensity, H?

A8) 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

Q9) Explain Wave Equation.

A9) 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.

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)

Q10) Explain Plane electromagnetic waves in vacuum.

A10) 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)

Q11) Explain transverse nature.

A11) 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.

Q12) Explain Poynting theorem.

A12) 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

Q13) Using Maxwell’s equations, how the energy density carried by the electromagnetic wave can be calculated.

A13) Using Maxwell’s equation,

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.