Electromagnetic Theory
Course Outlines
Introduction:
Electromagnetic theory is concerned with the study of charges at rest and in motion. Electromagnetic principles are fundamental to the study of electrical engineering. Electromagnetic theory is also required for the understanding, analysis and design of various electrical, electromechanical and electronic systems.
Electromagnetic theory can be thought of as generalization of circuit theory. Electromagnetic theory deals directly with the electric and magnetic field vectors where as circuit theory deals with the voltages and currents. Voltages and currents are integrated effects of electric and magnetic fields respectively.
Electromagnetic field problems involve three space variables along with the time variable and hence the solution tends to become correspondingly complex. Vector analysis is the required mathematical tool with which electromagnetic concepts can be conveniently expressed and best comprehended. Since use of vector analysis in the study of electromagnetic field theory is prerequisite, first we will go through vector algebra.
Applications of Electromagnetic theory:
This subject basically consist of static electric fields, static magnetic fields, time-varying fields & it’s applications.
One of the most common applications of electrostatic fields is the deflection of a charged particle such as an electron or proton in order to control it’s trajectory. The deflection is achieved by maintaining a potential difference between a pair of parallel plates. This principle is used in CROs, ink-jet printer etc. Electrostatic fields are also used for sorting of minerals for example in ore separation. Other applications are in electrostatic generator and electrostatic voltmeter.
The most common applications of static magnetic fields are in dc machines. Other applications include magnetic deflection, magnetic separator, cyclotron, hall effect sensors, magneto hydrodynamic generator etc.
1: Parallelogram law and
2: Head & tail rule
Scaling of a vector is defined as , where is scaled version of vector Some important laws of vector algebra are:
and is a scalar.
Commutative Law..........................................(1.3)
Associative Law.............................................(1.4)
Distributive Law ............................................(1.5)
The position vector of a point P is the directed distance from the origin (O) to P, i.e., =
.
If = OP and
= OQ are the position vectors of the points P and Q then the distance vector
Fig 1.3: Distance Vector
Product of Vectors
When two vectors and are multiplied, the result is either a scalar or a vector depending how the two vectors were multiplied. The two types of vector multiplication are:
Scalar product (or dot product) Vector product (or cross product)
gives a scalar. gives a vector.
The dot product between two vectors is defined as
= |A||B|cosθAB ..................(1.6)
Vector product
is unit vector perpendicular to and
Fig 1.4 : Vector dot product
The dot product is commutative i.e.,
Associative law does not apply to scalar product.
The vector or cross product of two vectors and
and distributive i.e.,
is denoted by .
.
is a vector
and direction is
perpendicular to the plane containing and given by right hand rule.
, the magnitude is given by
............................................................................................(1.7)
where
is the unit vector given by,
.
The following relations hold for vector product.
= i.e., cross product is non commutative ..........(1.8) i.e., cross product is distributive.......................(1.9)
i.e., cross product is non associative..............(1.10)
Scalar and vector triple product :
Scalar triple product
.................................(1.11)
Vector triple product
...................................(1.12)
Co-ordinate Systems:
In order to describe the spatial variations of the quantities, we require using appropriate co-ordinate system. A point or vector can be represented in an orthogonal coordinate system. An orthogonal system is one in which the co-ordinates are mutually perpendicular.
In electromagnetic theory many physical quantities are vectors, which are having different components. So we use orthogonal co-ordinate systems for representing those quantities and depending on the symmetry of the physical quantities different coordinate systems are used.
Cartesian Co-ordinate System :
A point P(x, y, z) in Cartesian co-ordinate system is represented as intersection of three
planes x = constant, y = constant and z = constant, as shown in the figure below. The unit vectors along the three axes are as shown in the figure.
Coordinate system represented by (x,y,z) that are three orthogonal vectors in straight lines that intersect at a single point (the origin). The range of variation along the three axes are shown below.
The vector A in this coordinate system can be written as,
The differential lengths, area and volumes are as shown below.
Cylindrical Co-ordinate System :
For cylindrical coordinate systems we have
as shown in figure below.
Figure: Cylindrical Coordinate System
Cylindrical Coordinate system represented by ( , ,z) that are three orthogonal vectors, varies in the range,
The vector A in this coordinate system can be written as,
The following equations can be used to convert between cylindrical and Cartesian coordinate systems,
The differential elements in cylindrical coordinate system are shown below.
Spherical co-ordinate system:
Coordinate system represented by (r, , ) that are three orthogonal vectors (as shown in the figure below) emanating from or revolving around the origin in the range,
The vector A in this coordinate system can be written as,
The following equations can be used to convert between spherical and Cartesian coordinate systems.
The unit vectors in the three orthogonal directions are,
z
The differential elements in spherical coordinate system are shown below.
Co-ordinate transformation:
Matrix Transformations: Cartesian to Cylindrical
Matrix Transformations: Cartesian to Spherical
Del operator:
Del is a vector differential operator. The del operator will be used in for differential operations throughout any course on field theory. The following equation is the del operator for different coordinate systems.
Gradient of a Scalar:
-Fundamental properties of the gradient of a scalar field
and is called the directional derivative of V along a. This is the rate of change of V in the direction of a.
Divergence of a Vector:
Divergence Theorem:
Curl of a Vector:
-Curl of a vector in each of the three primary coordinate systems are,
Stokes Theorem:
Classification of vector field:
The vector field, A, is said to be divergenceless ( or solenoidal) if .
Solved Examples:
1. Given that
Solution:
a. We know that for two given vector A and B,
For the two vectors A and B
or
b. We know that is perpendicular to both A and B.
=
The unit vector
perpendicular to both A and B is given by,
2. Given the vectors
Find :
Solution:
The vector B is cylindrical coordinates. This vector in Cartesian coordinate can be written as:
Where
The point P(0,2,-3) is in the y-z plane for which
.
a. C = A + B
=
=
b. Component of A along B is
where is the angle between A is and B.
i.e.,
=
3. A vector field is given by
Transform this vector into rectangular co-ordinates and calculate its magnitude at P(1,0,1).
Solution:
Given,
The components of the vector in Cartesian coordinates can be computed as follows:
6. A given vector function is defined by P1(1, 1, -1) to P2(2, 4, -1).
. Evaluate the scalar line integral from a point
Is F a conservative field?
Solution:
a. For evaluating the line integral along the parabola
, we find that
d y = 2 x d x
b. In this case we observe that z1 = z2 = -1, hence the line joining the points P1 and P2 lies in the z = -1 plane and can be represented by the equation
Or, y = 3x -2
d y = 3 d x
F . d l = (3x -2)d x + x . 3d x
= (6x -2)d x
=
= 7
The field F is a conservative field.
7. If
, calculate
over a hemispherical surface bounded by r =2 &
.
Solution:
In spherical polar coordinates
QUESTIONS:
(Preliminary Questions)
1.A small sphere of radius r and charge q is enclosed by a spherical shell of radius R and charge Q. Show that if q is positive, charhe q will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge Q on the shell is.(NCERT PHYSICS).
determine A,B and C.
r3 r3
r
E = (100 cos ) a + (50Sin ) a ;
Calculate the |E| and a unit vector in Cartesian coordinate in the
direction of E at a point (r=2, = 600 , =200)
Field:
A field is a function that specifies a particular physical quantity everywhere in a region. Depending upon the nature of the quantity under consideration, the field may be a vector or a scalar field. Example of scalar field is the electrostatic potential in a region while electric or magnetic fields at any point is the example of vector field.
Static Electric Fields:
Electrostatics can be defined as the study of electric charges at rest. Electric fields have their sources in electric charges. The fundamental & experimentally proved laws of electrostatics are Coulomb’s law & Gauss’s theorem.
Coulomb’s law & Electric field Intensity
Statement: The force between two point charges separated in vacuum or free space by a distance which is large compared to their size is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. It acts along the line joining the two charges.
Mathematically,
In SI units, Q1 and Q2 are expressed in Coulombs(C) and R is in meters.
Force F is in Newtons (N) and
, is called the permittivity of free space & it’s magnitude is
0
=8.854 x 10-12 =
36
109 F/m.
1
(We are assuming the charges are in free space. If the charges are any other dielectric medium, we
will use instead where is called the relative permittivity or the dielectric constant of the medium).
Therefore .......................(2.1)
As shown in the Figure 2.1 let the position vectors of the point charges Q1and Q2 are given by and
. Let represent the force on Q1 due to charge Q2.
Fig 2.1: Coulomb's Law
The charges are separated by a distance of
. We define the unit vectors as
and
..................................(2.2)
can be defined as
.
Similarly, the force on Q1 due to charge Q2 can be calculated and if
write .
represents this force then we can
Suppose a charge q is placed in the vicinity of three other charges, q1, q2, and q3, as is shown in Figure 2.2. Coulomb's law can be used to calculate the electric force between q and q1, between q and q2, and between q and q3. Experiments have shown that the total force exerted by q1, q2 and q3 on q is the vector sum of the individual forces, as shown in the equation below;
Figure 2.2. Superposition of electric forces.
Electric Field
The electric field intensity or the electric field strength at a point is defined as the force per unit charge. That is
or, .......................................(2.2)
The electric field intensity E at a point r (observation point) due a point charge Q located at is given by:
(source point)
..........................................(2.3)
For a collection of N point charges Q1 ,Q2 ,.........QN located at , ,...... , the electric field intensity at point is obtained as
........................................(2.4)
The expression (2.4) can be modified suitably to compute the electric filed due to a continuous distribution of charges.
For an elementary charge , i.e. considering this charge as point charge, we can write the field expression as:
.............(2.5)
When this expression is integrated over the source region, we get the electric field at the point P due to this distribution of charges. Thus the expression for the electric field at P can be written as:
..........................................(2.6)
Similar technique can be adopted when the charge distribution is in the form of a line charge density or a surface charge density.
........................................(2.7)
........................................(2.8)
Electric flux density:
As stated earlier electric field intensity or simply ‘Electric field' gives the strength of the field at a particular point. The electric field depends on the material media in which the field is being considered. The flux density vector is defined to be independent of the material media (as we'll see that it relates to the charge that is producing it).For a linear isotropic medium under consideration; the flux density vector is defined
as:
……………..(2.9)
We define the electric flux Y as
.....................................(2.10)
Gauss's Law: Gauss's law is one of the fundamental laws of electromagnetism and it states that the total electric flux through a closed surface is equal to the total charge enclosed by the surface.
Fig 2.3: Gauss's Law
Application of Gauss's Law
Gauss's law is particularly useful in computing or where the charge distribution has some symmetry. We shall illustrate the application of Gauss's Law with some examples.
1.An infinite line charge
Let’s consider the problem of determination of the electric field produced by an infinite line charge of density rLC/m. Let us consider a line charge positioned along the z-axis as shown in Fig. 2.4(a)
Since the line charge is assumed to be infinitely long, the electric field will be of the form as shown in Fig. 2.4(b).
If we consider a close cylindrical surface as shown in Fig. 2.4(a), using Gauss's theorem we can write,
.....................................(2.11)
Considering the fact that the unit normal vector to areas S1 and S3 are perpendicular to the electric field, the surface integrals for the top and bottom surfaces evaluates to zero.
Fig 2.4: Infinite Line Charge
Hence we can write,
.....................................(2.12)
2. Infinite Sheet of Charge
As a second example of application of Gauss's theorem, we consider an infinite charged sheet covering the x-
z plane as shown in figure 2.5.
Assuming a surface charge density of for the infinite surface charge, if we consider a cylindrical volume having sides placed symmetrically as shown in figure 2.5, we can write:
..............(2.13)
Fig 2.5: Infinite Sheet of Charge
It may be noted that the electric field strength is independent of distance. This is true for the infinite plane of charge; electric lines of force on either side of the charge will be perpendicular to the sheet and extend to infinity as parallel lines. As number of lines of force per unit area gives the strength of the field, the field becomes independent of distance. For a finite charge sheet, the field will be a function of distance.
Electrostatic Potential and Equipotential Surfaces
Electrostatic potential is related to the work done in carrying a charge from one point to the other in the
presence of an electric field. Let us suppose that we wish to move a positive test charge from a point P to another point Q as shown in the Fig.2.8.
Fig 2.8: Movement of Test Charge in Electric Field
The work done by this external agent in moving the charge by a distance is given by:
.............................(2.14)
The negative sign accounts for the fact that work is done on the system by the external agent.
.....................................(2.15)
The potential difference between two points P and Q , VPQ, is defined as the work done per unit charge, i.e.
...............................(2.16)
It may be noted that in moving a charge from the initial point to the final point if the potential difference is positive, there is a gain in potential energy in the movement, external agent performs the work against the field. If the sign of the potential difference is negative, work is done by the field.
We will see that the electrostatic system is conservative in that no net energy is exchanged if the test charge is moved about a closed path, i.e. returning to its initial position. Further, the potential difference between two points in an electrostatic field is a point function; it is independent of the path taken. The potential difference is measured in Joules/Coulomb which is referred to as Volts.
Considering the movement of a unit positive test charge from an arbitrary point B to another arbitrary point A ,
we can write an expression for the potential difference as:
..................................(2.17)
So, the potential difference is independent of the path taken as it only depends on the initial & final points. It is customary to choose the potential to be zero at infinity. Thus potential at any point ( rA = r) due to a point charge Q can be written as the amount of work done in bringing a unit positive charge from infinity to that point (i.e. rB = 0).
..................................(2.18)
Or, in other words,
..................................(2.19)
Let us now consider a situation where the point charge Q is not located at the origin as shown in Fig. 2.9.
Fig 2.9: Electrostatic Potential due a Displaced Charge
The potential at a point P becomes
..................................(2.20)
Let us first consider N point charges Q1, Q2,.....QN located at points with position vectors , ,...... . The potential at a point having position vector can be written as:
..................................(2.21a)
or,
...........................................................(2.21b)
For continuous charge distribution, we replace point charges Qn by corresponding charge elements
or or depending on whether the charge distribution is linear, surface or a volume charge distribution and the summation is replaced by an integral. With these modifications we can write:
For line charge,
..................................(2.22)
For surface charge,
.................................(2.23)
For volume charge, .................................(2.24)
It may be noted here that the primed coordinates represent the source coordinates and the unprimed coordinates represent field point.
Since the potential difference is independent of the paths taken, VAB = - VBA , and over a closed path,
.................................(2.25)
Applying Stokes's theorem, we can write:
............................(2.26)
from which it follows that for electrostatic field,
........................................(2.27)
Any vector field
that satisfies
is called an irrotational field.
From our definition of potential, we can write
.................................(2.28)
from which we obtain,
..........................................(2.29)
Electric Dipole
An electric dipole consists of two point charges of equal magnitude but of opposite sign and separated by a small distance.
As shown in figure 2.10, the dipole is formed by the two point charges Q and -Q separated by a distance d, the charges being placed symmetrically about the origin. Let us consider a point P at a distance r, where we are interested to find the field.
Fig 2.10 : Electric Dipole
The potential at P due to the dipole can be written as:
..........................(2.30)
When r1 and r2>>d, we can write Therefore,
and
.
....................................................(2.31)
We can write,
The quantity
...............................................(2.32)
is called the dipole moment of the electric dipole
Hence the expression for the electric potential can now be written as:
................................(2.33)
It may be noted that while potential of an isolated charge varies with distance as 1/r that of an electric dipole varies as 1/r2 with distance.
If the dipole is not centered at the origin, but the dipole center lies at , the expression for the potential can be written as:
........................(2.34)
The electric field for the dipole centered at the origin can be computed as
........................(2.35)
is the magnitude of the dipole moment. Once again we note that the electric field of electric dipole varies as 1/r3 where as that of a point charge varies as 1/r2.
Work Done by the Electrostatic Field
Work and Energy in Electrostatic Fields
If the charges were positioned in reverse order then:
Therefore one can write that the energy, W, present in an electrostatic field fue to different charge distributions is:
And since we can show by Gauss’ Law that:
, so,
QUESTIONS:
1.An electric field intensity is given as
r3 r3
r
E = (100 cos ) a + (50Sin ) a ;
Calculate the |E| and a unit vector in Cartesian coordinate
in the
direction of E at a point (r=2, = 600 , =200)
2.Derive the expression for electric field due to two equal point charges of opposite sign(Electric dipole). 3.Give the basic Concepts of transformation of one coordinate system to another
Ex =
s
20
[1 x(R2 + x2 )1/ 2 ]
SUMMARY:
MODULE-II
Currents & Conductors:
Convection and Conduction Currents:
Convection Current Density
Where u is the velocity vector of the fluid.
Conduction Current Density
–Current in a conductor
–Obeys Ohm’s law
Electrical resistivity:
So, we can write,
Continuity Equation
Applying Stokes Theorem,
The above equation is continuity equation.
QUESTIONS:
.A=- dV
dt
Coh1 (h / R) 2
where = Conductivity of the lake water.
3.State the Continuity equation for steady currents
Polarization in Dielectrics
Where,
Now,
Where the ’ operator is with respect to (x’,y’,z’).
So, we can define two charge densities,
When polarization occurs, an equivalent volume charge density, pv , is formed throughout the dielectric, while an equivalent surface charge density, ps , is formed over the surface.
Hence,
permittivity of free space from the electric susceptibility to make the units dimensionless. Thus we have
We know,
Thus, the dielectric constant (or relative permittivity) of the material, r , is the ratio of the permittivity to that of free space.
Electric field in material medium:
We have considered charge distributions only in free space & found it’s effect in terms of electric field intensity, electric flux density & electrostatic potential. Now we’ll consider effect of charge distributions in material medium.
In general, based on the electric properties, materials can be classified into three categories: conductors, semiconductors and insulators (dielectrics). In conductor , electrons in the outermost shells of the atoms are very loosely held and they migrate easily from one atom to the other. Most metals belong to this group. The electrons in the atoms of insulators or dielectrics remain confined to their orbits and under normal circumstances they are not liberated under the influence of an externally applied field. The electrical properties of semiconductors fall between those of conductors and insulators since semiconductors have very few numbers of free charges. The parameter conductivity is used characterizes the macroscopic electrical property of a material medium.
Conductors
If some free charge is introduced inside a conductor, the charges will experience a force due to mutual repulsion and owing to the fact that they are free to move, the charges will appear on the surface. The charges will redistribute themselves in such a manner that the field within the conductor is zero.
Therefore, under steady condition, inside a conductor , and using Gauss’s theorem
= 0 .......................(2.36)
, so a conductor behaves as an equipotential surface.
We know
Boundary conditions:
Boundary conditions govern the behavior of electric fields at the boundary (interface) between two
different media. The interface may be between two dielectrics or between a conductor & free space or between a conductor & dielectric. The last two cases will be special cases for first case.To complete this analysis we will use Gauss’s theorem & conservative nature of electrostatic fields.
Dielectric-Dielectric Boundary
Applying,
So,
Thus, tangential E undergoes no change and is continuous across the boundary condition Tangential D on the other hand is discontinuous across the interface.
Applying,
So,
Thus, normal D undergoes no change and is continuous across the boundary condition Normal E on the other hand is discontinuous across the interface.
So, we have,
Conductor-Dielectric Boundary
field inside the conductor) and a dielectric, 2 .
Law of Refraction:
Electrostatic Boundary Value Problems
Poisson’s and Laplace’s Equations for Electrostatics:
So,
The above equation is known as Poisson’s equation. For a charge-free region the above equation can be written as,
The above equation is known as Laplace’s equation.
Resistance:
For a uniform conductor, the resistance is given by,
Capacitance
The negative sign is dropped in the definition above because we are interested in the absolute value of the voltage drop.
Method of Images
QUESTIONS:
in parallel. Show that there is a loss of electrostatic energy amounting to
(C q C q )2
2C1C2 (C1 + C2 )
2 1 1 2 .
In
what form does this energy appear?
8.Compute the work done in moving a point charge Q around a closed loop in a static field. What is the
nature of electric field?
MODULE-III
Introduction to Magnetic Fields
Electrostatic fields are generated by static charges, magnetostatic fields are generated by static currents (charges that move with constant velocity in a particular direction).
Biot-Savart’s Law
Considering different current distributions(as shown above) we can rewrite expression for field intensity as below,
H Field From a Strait Current Carrying Filament
Now,
Using Biot-savart’s law we can find the expression for field intensity due to different current carrying conductor configurations.
Ampere’s law:
The line integral of H around a closed path is the same as the net current, Ienc, enclosed by the path,
We can also write using current density,
So, from the above two equations,
Applications of Ampere’s Circuit Law
Ampere’s Circuit Law: Infinite Sheet of Current
Thus for an infinite sheet of charge,
So, by using Ampere’s circuital law, the expressions for magnetic field intensity of different structures can be derived.
Magnetic Flux Density
–There are NO magnetic monopoles, stipulating that an isolated magnetic charge DOES NOT EXIST
–The minimum field requirement for magnetics is a dipole. So, mathematically,
Maxwell’s Eqns. for Static Fields
Magnetic Scalar & Vector Potential
We can define a magnetic field using the following requirements.
, we can define a magnetic scalar potential Vm related to H when the current density is
, We can define the Magnetic Vector Potential
QUESTIONS:
the opposite direction (as shown in Fig below) is given by,
z
ln( )a
1
2 r
A = 0 I r2
P
0 ni tan .
2a n
Show that this expression approaches the induction at the center of a circle as n is
indefinitely increased.
MODULE-III
Magnetic Forces Materials and Devices:
Lorentz Force Law:
Where ‘a’ is the acceleration of the particle is space.
Force on a Current Element
Thus we can solve for the force on the first wire:
From Biot- Savart’s law we have,
On substituting,
Using the above method we can find the force between two current carrying conductors.
Magnetic Torque and Moment:
And for a uniform magnetic field,
But,
,so
Where we can now define a quantity m as the magnetic dipole moment with units A/m2 which is the product of the current and area of the loop in the direction normal the surface area defined by the loop
QUESTIONS:
1.A wire is bent into a plane to form a square of 30cm side and a current of 10 A is passed through it. Calculate H at the centre of the loop.
Magnetization in Materials:
We know,
So,
Again,
So,
Where b in the J and K terms represents a bound current densities
For any linear magnetic material medium we have,
So,
Where is called the permeability of the material and is measured in H/m r is called the relative permeability.
Classification of Magnetic Materials:
Classification of Magnetic Materials
–Occurs when the magnetic fields in the material due to individual electron moments cancels each other out. Thus the permanent magnetic moment of each atom is zero.
–Such materials are very weakly affected by magnetic fields.
–Diamagnetic materials include Copper, Bismuth, silicon, diamond, and sodium chloride (table salt)
–In general this effect is temperature independent. Thus, for example, there is no technique for magnetizing copper
–Superconductors exhibit perfect diamagnetism. The effect is so strong that magnetic fields applied across a superconductor do not penetrate more than a few atomic layers, resulting in B=0 within the material
–Materials whose atoms exhibit a slight non-zero magnetic moment
–Paramangetism is temperature dependent
–Most materials (air, tungsten, potassium, monell) exhibit paramagnetic effects that provide slight magnetization in the presence of large fields at low temperatures
–Occurs in atoms with a relatively large magnetic moment
–Examples: Cobalt, Iron, Nickel, various alloys based on these three
–Capable of being magnetized very strongly by a magnetic field
–Retain a considerable amount of their magnetization when removed from the field
–Lose their ferromagnetic properties and become linear paramagnetic materials (non magnetic) when the temperature is raised above a critical temperature called the Curie temperature.
–Their magnetization is nonlinear. Thus the constitutive relation B= depends directly on B and cannot be represented by a single value.
0
r H does not hold because
r
Magnetic Boundary Conditions:
Inductors and Inductance:
Where L is a constant of proportionality called the inductance of the circuit. A circuit that contains inductance is said to be an inductor.
where L is measured in units of Henrys (H) = Wb/A.
Inductors and Inductance:
Magnetic Energy
We can derive a similar term as derived for electric energy, for magnetic energy using the relation for energy as a function of inductance.
Magnetic Circuits
Maxwell’s Equations:
Faraday’s Law for induced emf:
where, as before, is the flux linkage, is the magnetic flux, N is the number of turns in the inductor, and t represents a time interval. The negative sign shows that the induced voltage acts to oppose the flux producing it.
Transformer and Motional Electromotive Forces:
Displacement Current:
Based on the equation for displacement current density, we can define the displacement current in a circuit as shown. Applying Ampere’s circuit law to a closed path provides the following eqn. for current on the first side of the capacitive element. However surface 2 is the opposite side of the capacitor and has no conduction current allowing for no enclosed current at surface 2. If J =0 on the second surface then Jd must be generated on the second surface to create a time displaced current equal to current on surface 1.
If J =0 on the second surface then Jd must be generated on the second surface to create a time displaced current equal to current on surface 1.
We know,
So,
Maxwell’s Time Dependent Equations
The Maxwell’s equations for time dependent fields are,
QUESTIONS:
7.Show that the magnetic induction in Weber per square metre at the center of a square circuit of length l
on a side carrying a current i is
2
l
20i where i is in amperes and l is in meteres.
8.Write the expressions for vector magnetic potentials for three standard current configurations i.e. current filament, sheet current and volume current.
9.Derive Poisson’s equation and also its analogous one for static magnetic field.
MODULE-IV
Plane Wave:
A uniform plane wave is the wave that the electric field, E or magnetic field, H in same direction, same magnitude and same phase in infinite planes perpendicular to the direction of propagation. A plane wave
has no electric field, and magnetic field, components along its direction of propagation.
Wave Equations:
If the wave is in simple ( linear, isotropic and homogeneous ) nonconducting medium ( =0), Maxwell’s equation reduce to,
The first-order differential equations in the two variables E and H . They can combine to give E or H
alone using second-order equation.
Using Maxwell’s equation,
The curl of equation of (1)
Replacing in equation (2)
We know that
because of equation (3), thus the wave equation is
The wave equation also can written as
(a)
Assuming an implicit time dependence in the field vector. Equation (a) also called Helmholtz equation. The k is called the wave number or propagation constant.
where c is the velocity of light in free space.
For magnetic intensity domain, H , we have,
^
For a uniform plane wave with an electric field E = x Ex traveling in the z-direction, the wave equation can be reduced as
The solution of this wave equation,
Where is the attenuation constant of the medium and is its phase constant.
The associated magnetic field, H ,
where is the intrinsic impedance of the medium.
The k is called the wave number or propagation constant.
The wave number can also be written in terms of and .
Thus,
By solving (1) & (2),
So for different medium,
Electromagnetic Phenomena are described by using four Maxwell’s equations:
Poynting Vector and Power Flow in Electromagnetic Fields:
Electromagnetic waves can transport energy from one point to another point. The electric and magnetic field intensities associated with a travelling electromagnetic wave can be related to the rate of such energy transfer.
Let us consider Maxwell's Curl Equations:
Using vector identity
The above curl equations we can write
,
And .
In simple medium where
and are constant, we can write
Applying Divergence theorem we can write,
…………….(a)
The term
represents the rate of change of energy stored in the electric and
magnetic fields and the term represents the power dissipation within the volume. Hence right hand side of the equation (a) represents the total decrease in power within the volume under consideration.
The left hand side of equation (6.36) can be written as where
(W/mt2) is called the Poynting vector and it represents the power density vector associated with the electromagnetic field. The integration of the Poynting vector over any closed surface gives the net power flowing out of the surface. Equation (6.36) is referred to as Poynting theorem and it states that the net power flowing out of a given volume is equal to the time rate of decrease in the energy stored within the volume minus the conduction losses.
Poynting vector for the time harmonic case:
For time harmonic case, the time variation is of the form , and we have seen that instantaneous value
of a quantity is the real part of the product of a phasor quantity and when is used as reference. For example, if we consider the phasor
then we can write the instanteneous field as
when E0 is real.
Let us consider two instanteneous quantities A and B such that
,
where A and B are the phasor quantities. i.e,
Therefore,
, the time average value of the product form AB, denoted
Since A and B are periodic with period by can be written as
Further, considering the phasor quantities A and B, we find that
and
, where * denotes complex conjugate.
The poynting vector
can be expressed as
..............................(b)
If we consider a plane electromagnetic wave propagating in +z direction and has only from (b) we can write:
component,
Using (6.41)
, for the plane wave under consideration.
where and
For a general case, we can write
We can define a complex Poynting vector
and time average of the instantaneous Poynting vector is given by .
Polarisation of plane wave:
The polarization of a plane wave can be defined as the orientation of the electric field vector as a function of time at a fixed point in space. For an electromagnetic wave, the specification of the orientation of the electric field is sufficient as the magnetic field components are related to electric field vector by the Maxwell's equations.
Let us consider a plane wave travelling in the +z direction. The wave has both Ex and Ey components.
The corresponding magnetic fields are given by,
Depending upon the values of Eox and Eoy we can have several possibilities:
with the axis of polarisation inclined at an angle fig 6.4.
, with respect to the x-axis. This is shown in
Fig 6.4 : Linear Polarisation
If Eox and Eoy are complex with different phase angles, will not point to a single spatial direction. This is explained as follows:
Let ,
Then,
and ....................................(c)
To keep the things simple, let us consider a =0 and field on the z =0 plain.
From equation (c) we find that,
. Further, let us study the nature of the electric
and the electric field vector at z = 0 can be written as
………(d)
Assuming
, the plot of
for various values of t is hown in figure 6.5.
Figure 6.5 : Plot of E(o,t)
From equation (d) and figure (6.5) we observe that the tip of the arrow representing electric field vector traces an ellipse and the field is said to be elliptically polarized.
Figure 6.6: Polarisation ellipse
The polarisation ellipse shown in figure 6.6 is defined by its axial ratio(M/N, the ratio of semimajor to
semiminor axis), tilt angle (orientation with respect to xaxis) and sense of rotation(i.e., CW or CCW). Linear polarisation can be treated as a special case of elliptical polarisation, for which the axial ratio is infinite.
In our example, if , from equation (6.47), the tip of the arrow representing electric field vector traces out a circle. Such a case is referred to as Circular Polarisation. For circular polarisation the axial ratio is unity.
Figure 6.7: Circular Polarisation (RHCP)
Further, the circular polarisation is aside to be right handed circular polarisation (RHCP) if the electric field vector rotates in the direction of the fingers of the right hand when the thumb points in the direction of propagation-(same and CCW). If the electric field vector rotates in the opposite direction, the polarisation is asid to be left hand circular polarisation (LHCP) (same as CW).
In AM radio broadcast, the radiated electromagnetic wave is linearly polarised with the field vertical to the ground( vertical polarisation) where as TV signals are horizontally polarised waves. FM broadcast is usually carried out using circularly polarised waves.
In radio communication, different information signals can be transmitted at the same frequency at
orthogonal polarisation ( one signal as vertically polarised other horizontally polarised or one as RHCP while the other as LHCP) to increase capacity. Otherwise, same signal can be transmitted at orthogonal polarisation to obtain diversity gain to improve reliability of transmission.
Behaviour of Plane waves at the inteface of two media:
We have considered the propagation of uniform plane waves in an unbounded homogeneous medium. In
practice, the wave will propagate in bounded regions where several values of will be present. When plane wave travelling in one medium meets a different medium, it is partly reflected and partly transmitted. In this section, we consider wave reflection and transmission at planar boundary between two media.
Fig 6.8 : Normal Incidence at a plane boundary
Case1: Let z = 0 plane represent the interface between two media. Medium 1 is characterised
by and medium 2 is characterized by .
Let the subscripts 'i' denotes incident, 'r' denotes reflected and 't' denotes transmitted field components respectively.
The incident wave is assumed to be a plane wave polarized along x and travelling in medium 1 along direction. From equation (6.24) we can write
..................(e)
......................(f)
where and .
Because of the presence of the second medium at z =0, the incident wave will undergo partial reflection
and partial transmission. The reflected wave will travel along in medium 1. The reflected field components are:
...............................................(g)
.........(h)
The transmitted wave will travel in medium 2 along for which the field components are
............................................(i)
............................................(j)
where
In medium 1,
and
and
and in medium 2,
and
Applying boundary conditions at the interface z = 0, i.e., continuity of tangential field components and noting that incident, reflected and transmitted field components are tangential at the boundary, we can write
&
From equation (e) to (j) we get,
................................................................(k)
..............................................................(l)
Eliminating Eto ,
or,
or,
...............(m)
is called the reflection coefficient. From equation (k) & (l), we can write
or,
is called the transmission coefficient.
We observe that,
The following may be noted
(i) both and T are dimensionless and may be complex
(ii)
Let us now consider specific cases:
Case I: Normal incidence on a plane conducting boundary
The medium 1 is perfect dielectric and medium 2 is perfectly conducting
.
From (k) and (l)
= -1
and T =0
Hence the wave is not transmitted to medium 2, it gets reflected entirely from the interface to the medium 1.
&
Proceeding in the same manner for the magnetic field in region 1, we can show that,
The wave in medium 1 thus becomes a standing wave due to the super position of a forward travelling
wave and a backward travelling wave. For a given ' t', both and vary sinusoidally with distance measured from z = 0. This is shown in figure 6.9.
Figure 6.9: Generation of standing wave
Zeroes of E1(z,t) and
Maxima ofH1(z,t).
Maxima of E1(z,t) and
zeroes ofH1(z,t).
Case2: Normal incidence on a plane dielectric boundary
If the medium 2 is not a perfect conductor (i.e. ) partial reflection will result. There will be a reflected wave in the medium 1 and a transmitted wave in the medium 2.Because of the reflected wave, standing wave is formed in medium 1.
From above equations we can write
Let us consider the scenario when both the media are dissipation less i.e. perfect dielectrics ( )
In this case both
and become real numbers.
..................(n)
From (n), we can see that, in medium 1 we have a traveling wave component with amplitude TEio and a standing wave component with amplitude 2JEio.
The location of the maximum and the minimum of the electric and magnetic field components in the
medium 1from the interface can be found as follows. The electric field in medium 1 can be written as
If i.e. >0
The maximum value of the electric field is
and this occurs when
or
, n = 0, 1, 2, 3.......................(o)
The minimum value of
is
.................(p)
And this occurs when
or
, n = 0, 1, 2, 3.............................(q)
For i.e. <0
The maximum value of
is which occurs at the zmin locations and the minimum value
of is
which occurs at zmax locations as given by the equations (o) and (q).
From our discussions so far we observe that
can be written as
The quantity S is called as the standing wave ratio.
As the range of S is given by
We can write the expression for the magnetic field in medium 1 as
From above equation we can see that will be maximum at locations where vice versa.
In medium 2, the transmitted wave propagates in the + z direction.
is minimum and
Oblique Incidence of EM wave at an interface
So far we have discuss the case of normal incidence where electromagnetic wave traveling in a lossless medium impinges normally at the interface of a second medium. In this section we shall consider the case of oblique incidence. As before, we consider two cases
i.
ii.
When the second medium is a perfect conductor. When the second medium is a perfect dielectric.
A plane incidence is defined as the plane containing the vector indicating the direction of propagation of the incident wave and normal to the interface. We study two specific cases when the incident electric
field is perpendicular to the plane of incidence (perpendicular polarization) and is parallel to the
plane of incidence (parallel polarization). For a general case, the incident wave may have arbitrary polarization but the same can be expressed as a linear combination of these two individual cases.
Oblique Incidence at a plane conducting boundary
i.
Perpendicular Polarization
The situation is depicted in figure 6.10.
Figure 6.10: Perpendicular Polarization
As the EM field inside the perfect conductor is zero, the interface reflects the incident plane wave.
and respectively represent the unit vector in the direction of propagation of the incident and reflected waves, is the angle of incidence and is the angle of reflection.
We find that
Since the incident wave is considered to be perpendicular to the plane of incidence, which for the present case happens to be xz plane, the electric field has only y-component.
Therefore,
The corresponding magnetic field is given by
Similarly, we can write the reflected waves as
Since at the interface z=o, the tangential electric field is zero.
The above equation is satisfied if we have
The condition
is Snell's law of reflection.
The total electric field is given by
Similarly, total magnetic field is given by
From above two equations we observe that
1. Along z direction i.e. normal to the boundary
y component of to and
component of
and x component of where
and x component of
maintain standing wave patterns according
. No average power propagates along z as y are out of phase.
2. Along x i.e. parallel to the interface
y component of and z component of are in phase (both time and space) and propagate with phase velocity
The wave propagating along the x direction has its amplitude varying with z and hence constitutes a non
uniformplane wave. Further, only electric field is perpendicular to the direction of propagation (i.e. x), the magnetic field has component along the direction of propagation. Such waves are called transverse electric or TE waves.
ii. Parallel Polarization:
In this case also and are given by the derived equations. Here and component.
have only y
Figure 6.11: Parallel Polarization
With reference to fig (6.11), the field components can be written as:
Incident field components:
............................(r)
Reflected field components:
Since the total tangential electric field component at the interface is zero.
Which leads to and as before.
Substituting these quantities in (r) and adding the incident and reflected electric and magnetic field components the total electric and magnetic fields can be written as
Once again, we find a standing wave pattern along z for the x and y components of
and , while a
non uniform plane wave propagates along x with a phase velocity given by
where . Since, for this propagating wave, magnetic field is in transverse direction, such waves are called transverse magnetic or TM waves.
Oblique incidence at a plane dielectric interface
We continue our discussion on the behavior of plane waves at an interface; this time we consider a plane dielectric interface. As earlier, we consider the two specific cases, namely parallel and perpendicular polarization.
Fig 6.12: Oblique incidence at a plane dielectric interface
For the case of a plane dielectric interface, an incident wave will be reflected partially and transmitted partially.
In Fig(6.12), corresponds respectively to the angle of incidence, reflection and transmission.
1. Parallel Polarization
As discussed previously, the incident and reflected field components can be written as
In terms of the reflection coefficient
The transmitted filed can be written in terms of the transmission coefficient T
We can now enforce the continuity of tangential field components at the boundary i.e. z=0
..........................(s)
are to be continuous at z=0 for all x , then form the phase matching we have
If both and
We find that
..........................(t)
Further, from equations (s) and (t) we have
or
..........................(u)
for which
From equation (u) we find that there exists specific angle
= 0 such that
or
Further,
For non magnetic material Using this condition
.........................(v)
rom equation (v), solving for
we get
This angle of incidence for which
= 0 is called Brewster angle. Since we are dealing with parallel
polarization we represent this angle by
so that
2. Perpendicular Polarization
For this case
Using continuity of field components at z=0
As in the previous case
Using these conditions we can write
.........................(w)
From equation (w) the reflection and transmission coefficients for the perpendicular polarization can be computed as
We observe that if
= 0 for an angle of incidence
Again
or
or
or
We observe if
.........................(x)
i.e. in this case of non magnetic material Brewster angle does not exist as
the denominator or equation (x) becomes zero. Thus for perpendicular polarization in dielectric media, there is Brewster angle so that can be made equal to zero.
From our previous discussion we observe that for both polarizations
from 1
Wave equations:
The Maxwell's equations in the differential form are
Let us consider a source free uniform medium having dielectric constant
, magneticpermeability
and conductivity . The above set of equations can be written as
Using the
vector identity , We can write
from 2
Substituting
But in source fr
ee(
In the same manner for equation eqn 1
) medium (eq3)
Since
from eqn 4, we can write
These two equations are known as wave equations.
Uniform plane waves:
A uniform plane wave is a particular solution of Maxwell's equation assuming electric field (and magnetic field) has same magnitude and phase in infinite planes
perpendicular to the direction of propagation..Since the plane wave will have no variation
along the plane perpendicular to zi.e., xy plane, . The Helmholtz's equation reduces to, The solution to this equation can be written as
are the amplitude constants (can be determined from boundary conditions).
In the time domain,
Figure : Plane wave traveling in the + z direction
As can be seen from the figure, at successive times, the wave travels in the +z direction.
If we fix our attention on a particular point or phase on the wave (as shown by the dot) i.e.
=
=
The intrinsic impedance of the free space. In the time domain,
Which represents the magnetic field of the wave traveling in the +z direction.For the negative traveling wave,
For the plane waves described, both the E & H fields are perpendicular to the direction of propagation, and these waves are called TEM (transverse electromagnetic) waves.