emt notes opt.unlocked

7/30/2019 Emt Notes Opt.unlocked

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Subject Name : ELECTROMAGNETIC

THEORY

Code : EE2202

Year : II Semester : III

Degree & Branch : B.E.-E.E.E. Section : A & B

UNIT 1 INTRODUCTION

Sources of EMF:

EMT NOTES
http://www.eeecube.blogspot.com/


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Types: Positive, Negative, Zero

Curl:

It is defined as the limit of its surface integral of its cross product with normal over a closed

surface per unit volume shrinks to zero.

Divergence Theorem:

Surface integral of normal component of any vector field is equal to volume integral of

divergence of that vector field over the volume V enclosing the surface S.

=sv

dsFDivFdv .

Stokes Theorem (or) Greens Theorem:

Line integral of any vector field F over the contour C is equal to surface integral of curl of

the vector field F over the surface S having a contour C.

=s c

dlFdscurlF ..

Co-ordinate systems:

Cartesian co-ordinate systems (x, y, z):


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Cylindrical co-ordinate systems (, , z):

Spherical co-ordinate systems(r, , ):


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Exercise:

1. Determine the volume of a cylinder of radius a m and height h m using differential volume in

cylindrical co-ordinates.

2. Determine the volume of a sphere of radius a m from the differential volume in spherical co-

ordinate systems.

UNIT 2 ELECTROSTATICS


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It is the branch of electromagnetic field dealing with the effects of electric charge at rest.

Coulombs law :

The force of attraction or repulsion between any two point charges is directly

proportional to the product of two charges and inversely proportional to square of distance

between them.

Types of charges :

Point.

Line.

Surface.

Volume.

Electric Field intensity :
http://www.eeecube.blogspot.com/http://en.wikipedia.org/wiki/Image:CoulombsLaw.svg


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The electric field is defined as the force (in Nw) per unit charge (in C). From this definition and

Coulomb's law, it follows that the magnitude of the electric fieldEcreated by a single point charge Q is

Electric Field due to a Point Charge q :

Magnitude is given by

Radially outwards for positive charges and radially inward for negative charges.

Electric Field due to a Infinite Line Charge with Uniform Charge Density

Magnitude:
http://en.wikipedia.org/wiki/Electric_fieldhttp://www.eeecube.blogspot.com/http://en.wikipedia.org/wiki/Image:EfieldTwoOppositePointCharges.svghttp://en.wikipedia.org/wiki/Electric_field


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Radially outwards for a positive perpendicular to the line charges.

It is assumed that the charges on the line are fixed in position. A conduction rod would produce just an

E-field, but as soon as you put any charge near the rod, the distribution of charge on the rod would

change.

Electric Field on the Axis of a Charged Ring with a Uniform Charge Density

Q = Total Charge on the Ring of radius R

Magnitude:

Outwards away from the ring along the axis of the ring if Q is positive.

The ring is assumed to lie in zy-plane with its axis along the x-axis


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The E-field (as a function of its location) at points other than along the axis is very complex and

it can not easily be expressed by such a simple function.

Electric Field due to Infinitely Flat, Charged-Plane with a Uniform Charge Density

Magnitude:

Outwards perpendicular to the plane.

For an infinite sheet the E-field is the same at any distance from the sheet, i.e. the E-field is

constant.

This expression is useful for electrical conductors because the charge spreads out on the

conductor's surface forming a surface charge distribution . Although the charge density is

not necessarily constant over the surface, but if you get close enough, the conductor's surface

will look like a infinitely charge sheet. The electric field will be perpendicular to the surface

and equal to half of a charged sheet because the E-field inside the conductor is zero.

Gausss law :
http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Gauss/IdealCond.htmlhttp://www.eeecube.blogspot.com/http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Gauss/IdealCond.html


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Gauss' law states that "the total electric flux through a closed surface is proportional to the total electric

charge enclosed within the surface". The constant of proportionality is thepermittivity of free space.

Mathematically, Gauss's law takes the form of an integral equation:

Alternatively, in differential form, the equation becomes

Applications of Gausss law:

1. Electric field due to an infinite charged wire

2. Electric field due to an infinite charged Sheet.

3. Electric field due to a spherical shell.

4. Electric field due to a spherical body.

Electrical Potential:

At a point in space, the electric potential is the potential energy per unit of charge that is

associated with a static (time-invariant) electric field. It is typically measured in volts, and is a Lorentz

scalarquantity. The difference in electrical potential between two points is known as voltage.

Mathematically, it is the potential (a scalar field) associated with the conservative electric field (

) that occurs when the magnetic field is time invariant (so that from

Faraday's law of induction).

Electric Field:
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In physics, the space surrounding an electric charge or in the presence of a time-varying

magnetic field has a property called an electric field (that can also be equated to electric flux density).

The electric field is a vector field with SI units of Nw/C (N C1) or, equivalently, volts permeter(V

m1).

A stationary charged particle in an electric field experiences a forceproportional to its charge

given by the equation,

Electric Field in free space:

Electric field is defined as the electric force per unit charge. The direction of the field is taken to

be the direction of the force it would exert on a positive test charge. The electric field is radially

outward from a positive charge and radially in toward a negative point charge.

CONDITIONS AT A BOUNDARY BETWEEN DIELECTRICS

Consider the interface between two dielectrics with different dielectrics. Let the two media 1 and 2have permittivities

1= 0 r1

2=0 r2The normal components of displacement densities are equal at the boundary between two dielectric

media.

Dn1= Dn2Tangential components of the fields E1 and E2 at the boundary are equal.

Et1= Et2

1 1

2 2

tan

tan

r

r

=

The angles 1 and 2 are the angles of incidence and refraction respectively.

Poissons and Lap laces equation:

Laplace's equation is apartial differential equation named afterPierre-Simon Laplace who first

studied its properties.
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In three dimensions, the problem is to find twice-differentiable real-valued functions, of real

variables,x,y, andz, such that

This is often written as or

where div is the divergence, and grad is the gradient,

or where is the Laplace operator.

In mathematics, Poisson's equation is a partial differential equation with broad utility in

electrostatics, mechanical engineering and theoretical physics. It is named after the French

mathematician,geometerandphysicistSimon-Denis Poisson. The Poisson equation is

where is the Laplace operator, andfand are real orcomplex-valued functions on a manifold. When

the manifold is Euclidean space, the Laplace operator is often denoted as and so Poisson's equation

is frequently written as

In three-dimensional Cartesian coordinates, it takes the form

Capacitance:

Capacitance is a measure of the amount of electric charge stored (or separated) for a given electric

potential. The most common form of charge storage device is a two-plate capacitor. If the charges on
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the plates are +Q and Q, and V give the voltage difference between the plates, then the capacitance is

given by

The SI unit of capacitance is the farad; 1 farad = 1 coulomb pervolt.

Where

C=Capacitance in farads

Q=Charge in coulomb on each conductor

V=Potential difference between the conductors(i) The capacitance between parallel plate capacitor which are separated by a distance

t,having surface area A and having dielectric of permittivity is

= 0 r.C= 0 rA/t

(ii) The capacitance of an isolated sphere

C=4 0r1(iii) The capacitance between two concentric spherical shells

C=4 0(ab/b-a)Where

a = radius of inner spherical shell

b = radius of outer spherical shell

(iv) The capacitance between co-axial cylinders

2

lnab

Cb

a

=

Wherea = radius of inner cylinder

b = radius of outer cylinder

Energy density:

Energy density is the amount ofenergy stored in a given system or region of space per unit volume.

Electric and magnetic fields store energy. In a vacuum, the (volumetric) energy density (in SI units) is

given by
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,

where E is the electric field and B is the magnetic field.

ENERGY DENSITY

Consider an elementary volume v with faces parallel to the capacitor plates (normal to the field E).If

the volume element is a cube with side t,the capacitance

C= (t)2/ t

= tV=E(t)

Energy stored in the element

2

2

1

( )( )2

1( )

2

W C V

W E v

=

=

The energy density w at any point

0

2

lim

1

2

1.

2

v

Ww

v

w E

w D E

=

=

=

Dielectric strength:

Dielectric strength of an insulating material, the maximum electric field strength that it can

withstand intrinsically without breaking down, i.e., without experiencing failure of its insulating

properties. Dielectric strength for a given configuration of dielectric material and electrodes, the

minimum electric field that produces breakdown.
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The theoretical dielectric strength of a material is an intrinsic property of the bulk material and is

dependent on the configuration of the material or the electrodes with which the field is applied. At

breakdown, the electric field frees bound electrons. If the applied electric field is sufficiently high, free

electrons may become accelerated to velocities that can liberate additional electrons during collisions

with neutral atoms or molecules in a process called avalanche breakdown. Breakdown occurs quite

abruptly (typically in nanoseconds)., resulting in the formation of an electrically conductive path and a

disruptive discharge through the material. For solid materials, a breakdown event severely degrades, or

even destroys, its insulating capability.

UNIT -3 MAGNETOSTATICS

Lorentz law of force:

The Lorentz force is the force on a point charge due to electromagnetic fields. It is given by the

following equation in terms of the electric and magnetic fields:[1]

Where, F is the force (in Nw) , E is the electric field (in V/m), B is the magnetic field (in T), q is the

electric charge of the particle (in coulombs), v is the instantaneous velocity of the particle (in meters

persecond). Or equivalently the following equation in terms of thevector potential and scalar potential:

Where, magnetic vector potential and electrostatic potential are related to E and B by
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Magnetic Field intensity:

Magnetic field strength (H) is the amount of magnetizing force. It is proportional to the length

of a conductor and the amount of electrical current passing through the conductor. Magnetic field

strength is a vector quantity whose magnitude is the strength of a magnetic field at a point in the

direction of the magnetic field at that point. Flux density (B), the amount of magnetism induced in a

body, is a function of the magnetizing force (H).

Biot-Savart law:

The BiotSavart Law is an equation in electromagnetism that describes the magnetic field B

generated by an electric current. The vector field B depends on the magnitude, direction, length, and

proximity of the electric current, and also on a fundamental constant called the magnetic constant. The

law is valid in the magnetostatic approximation.

The BiotSavart law is used to compute the magnetic field generated by asteady current, i.e. a

continual flow ofcharges, for example through a wire, which is constant in time and in which charge isneither building up nor depleting at any point. The equation is as follows:

Where,

is the current,

is a vector, whose magnitude is the length of the differential element of the wire, and whose

direction is the direction ofconventional current,

is the differential contribution to the magnetic field resulting from this differential element of

wire,

is the magnetic constant,
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is the displacement unit vector in the direction pointing from the wire element towards the

point at which the field is being computed,

and is the distance from the wire element to the point at which the field is being computed.

1. The field intensity H at the centre of a circular wire of radius a carrying a current I

2

IH k

a=

2. The field intensity H at a point P due to a straight conductor carrying a current I

1 2

(cos cos )4

P

IH k

h

= +

If the conductor is infinitely long, then the field intensity H at a point P

2

p

IH

h

=

3. The field intensity H at any point P on the line through the centre at a distance h from

the centre and perpendicular to the plane of a plane circular loop of radius a and

carrying current I

2

3

2 2 22( )

p

IaH

a h

=

+

Ifh=o, p coincides with O (centre of the wire loop)

2

IHa

=

4. The field intensity at any point along the axis of a Solenoid (Uniformly cylindrical coil

wound on a non-magnetic frame). Solenoid have N turns uniformly distributed over a

length and mean radius of the coil=a.

1 2(cos cos2

P

INH

L=

(a). Let P be at one of the ends of the solenoid.

2

PINH

L=

(b). Let P be at the centre of the solenoid.

P

INH

L=
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(b). Let P be point on the axis, mid way between one end and centre of the solenoid.

1

(1 co s2

P

INH

L= +

FORCE BETWEEN CURRENT CARRYING WIRES

Consider two straight, long parallel current carrying conductors placed D metre apart. Conductor 1

produces a field around and its value B1 at the location of conductor 2

0 11

2

IB

D

=

Conductor 2 carries a current I2 and is situated in a field with flux density B1

dF I dlX B=For a length of 1 metre, the force on the conductor 2

2 2 1

2 2 1

2 2 1

1

1(

(1 )

F I XB

F kI X jB

F i I B

=

=

=

The force of attraction

2 1

0 12

0 1 2

1

1 ( )2

2

F I B

IF I

D

I IF l

D

=

=

=

Force of attraction between two infinitely long parallel conductors per metre length is

0 1 2

2

I IF Nw

D

=

Field produced at conductor 1 due to the conductor 2

0 22

0 1 21 2

2

2

IB

D

I IF F

D

=

= =

Amperes law:
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Ampre's circuital law, discovered by Andr-Marie Ampre, relates the integrated magnetic field

around a closed loop to the electric current passing through the loop. It is the magnetic analogue of

Gauss's law, and one of the fourMaxwell's equations that form the basis ofclassical electromagnetism.

Integral form

The "integral form" of the original Ampre's Circuital law is:

Differential form

This equation can also be written in a "differential form". Again, this equation only applies in the case

where the electric field is constant in time. The equation states:

Magnetic flux density

A vector quantity measuring the strength and direction of the magnetic field around a magnet or an

electric current. Magnetic flux density is equal to magnetic field strength times the magnetic

permeability in the region in which the field exists. Electric charges moving through a magnetic field

are subject to a force described by the equation F = qv B, where q is the amount of electric charge, v

is the velocity of the charge, B is the magnetic flux density at the position of the charge, and is thevector product. Magnetic flux density also can be understood as the density of magnetic lines of force,

or magnetic flux lines, passing through a specific area. It is measured in units of tesla. Also called

magnetic flux, magnetic induction.
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UNIT 4 ELECTRODYNAMIC FIELDS

Faradays laws:

The law states that the induced electromotive force or EMF in any closed circuit is equal to the time

rate of change of the magnetic flux through the circuit.

Quantitatively, the law takes the following form:

.

Where, is the electromotive force (EMF) in volts

B is the magnetic flux through the circuit (in webers).

The direction of the electromotive force (the negative sign in the above formula) is given by Lenz's

law.

Induced emf:

For a given device, if a charge Q passes through that device, and gains an energy U, the net emf

for that device is the energy gained per unit charge, or U/Q. This has units ofvolts, or newton meters

per coulomb.

If the vector field f represents the force per unit charge on a charge carrier, the emf around a circuit Cis
http://en.wikipedia.org/wiki/Electromotive_forcehttp://en.wikipedia.org/wiki/Magnetic_fluxhttp://en.wikipedia.org/wiki/Electromotive_forcehttp://en.wikipedia.org/wiki/Volthttp://en.wikipedia.org/wiki/Magnetic_fluxhttp://en.wikipedia.org/wiki/Weber_(Wb)http://en.wikipedia.org/wiki/Lenz's_lawhttp://en.wikipedia.org/wiki/Lenz's_lawhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Chargehttp://en.wikipedia.org/wiki/Volthttp://en.wikipedia.org/wiki/Charge_carrierhttp://www.eeecube.blogspot.com/http://en.wikipedia.org/wiki/Electromotive_forcehttp://en.wikipedia.org/wiki/Magnetic_fluxhttp://en.wikipedia.org/wiki/Electromotive_forcehttp://en.wikipedia.org/wiki/Volthttp://en.wikipedia.org/wiki/Magnetic_fluxhttp://en.wikipedia.org/wiki/Weber_(Wb)http://en.wikipedia.org/wiki/Lenz's_lawhttp://en.wikipedia.org/wiki/Lenz's_lawhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Chargehttp://en.wikipedia.org/wiki/Volthttp://en.wikipedia.org/wiki/Charge_carrier


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Motional EMF:

When a conductor moves through a magnetic field an emf is produced in the conductor. The charges in

the conductor are carried along with the moving conductor and thus experience a magnetic force acting

upon them which causes them to move inside the conductor. As the conduction charges pile up at the

end of conductor creating an electric field in the cunductor. The conduction electrons will stop piling

up when the electric force on the interior conduction charges is equal to the magnetic force on those

same charges so that the net force on the conduction charges is zero.

Maxwells equation:

Maxwell's equations are a set of fourpartial differential equations that describe the properties of

the electric and magnetic fields and relate them to their sources, charge density and current density.

Individually, the equations are known as Gauss' law, Gauss' law for magnetism, Faraday's law

of induction, and Ampre's law with Maxwell's correction.

Name Differential form Integral form

Gauss' law:
http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/MagneticField/MFonCharge.htmlhttp://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/MagneticField/MFonCharge.htmlhttp://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/ElectricForce/ForceField.htmlhttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Charge_densityhttp://en.wikipedia.org/wiki/Current_densityhttp://en.wikipedia.org/wiki/Gauss'_lawhttp://en.wikipedia.org/wiki/Gauss'_law_for_magnetismhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_lawhttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Gauss'_lawhttp://www.eeecube.blogspot.com/http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/MagneticField/MFonCharge.htmlhttp://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/MagneticField/MFonCharge.htmlhttp://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/ElectricForce/ForceField.htmlhttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Charge_densityhttp://en.wikipedia.org/wiki/Current_densityhttp://en.wikipedia.org/wiki/Gauss'_lawhttp://en.wikipedia.org/wiki/Gauss'_law_for_magnetismhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_lawhttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Gauss'_law


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Gauss' law for magnetism:

Maxwell-Faraday equation

(Faraday's law of

induction):

Ampre's circuital law

(with Maxwell's

correction):

Displacement current:

The displacement current was introduced by Maxwell as the rate of change of the electric

displacement, D:

where D is the electric displacement field that enters Maxwell's equations. The electric displacement

field is defined as:
http://en.wikipedia.org/wiki/Gauss'_law_for_magnetismhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_lawhttp://en.wikipedia.org/wiki/Electric_displacementhttp://en.wikipedia.org/wiki/Electric_displacementhttp://en.wikipedia.org/wiki/Electric_displacement_fieldhttp://en.wikipedia.org/wiki/Maxwell's_equationshttp://en.wikipedia.org/wiki/Electric_displacement_fieldhttp://en.wikipedia.org/wiki/Electric_displacement_fieldhttp://www.eeecube.blogspot.com/http://en.wikipedia.org/wiki/Gauss'_law_for_magnetismhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Faraday's_law_of_inductionhttp://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_lawhttp://en.wikipedia.org/wiki/Electric_displacementhttp://en.wikipedia.org/wiki/Electric_displacementhttp://en.wikipedia.org/wiki/Electric_displacement_fieldhttp://en.wikipedia.org/wiki/Maxwell's_equationshttp://en.wikipedia.org/wiki/Electric_displacement_fieldhttp://en.wikipedia.org/wiki/Electric_displacement_field


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UNIT -5 ELECTROMAGNETIC WAVES

Electromagnetic radiation is produced whenever a charged particle, such as an electron, changes

its velocityi.e., whenever it is accelerated or decelerated. The energy of the electromagnetic radiation

thus produced comes from the charged particle and is therefore lost by it.

Electromagnetic wave equation:

The electromagnetic wave equation is a second-orderpartial differential equation that describes

the propagation ofelectromagnetic waves through a medium or in a vacuum. The homogeneous form

of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

Where c is the speed of light in the medium. In a vacuum, c = c0 = 299,792,458 meters per second,

which is the speed of light in free space. The electromagnetic wave equation derives from Maxwell's

equations.

Poynting vector:

The Poynting vector can be thought of as representing the energy flux (W/m2) of an

electromagnetic field. It is defined forfree space as
http://www.britannica.com/eb/article-9032271/electric-chargehttp://www.britannica.com/eb/article-9003469/accelerationhttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Electromagnetic_wavehttp://en.wikipedia.org/wiki/Medium_(optics)http://en.wikipedia.org/wiki/Vacuumhttp://en.wikipedia.org/wiki/Homogeneous_(mathematics)http://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Free_spacehttp://en.wikipedia.org/wiki/Maxwell's_equationshttp://en.wikipedia.org/wiki/Maxwell's_equationshttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Fluxhttp://en.wikipedia.org/wiki/Electromagnetic_fieldhttp://en.wikipedia.org/wiki/Free_spacehttp://www.eeecube.blogspot.com/http://www.britannica.com/eb/article-9032271/electric-chargehttp://www.britannica.com/eb/article-9003469/accelerationhttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Electromagnetic_wavehttp://en.wikipedia.org/wiki/Medium_(optics)http://en.wikipedia.org/wiki/Vacuumhttp://en.wikipedia.org/wiki/Homogeneous_(mathematics)http://en.wikipedia.org/wiki/Electric_fieldhttp://en.wikipedia.org/wiki/Magnetic_fieldhttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Free_spacehttp://en.wikipedia.org/wiki/Maxwell's_equationshttp://en.wikipedia.org/wiki/Maxwell's_equationshttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Fluxhttp://en.wikipedia.org/wiki/Electromagnetic_fieldhttp://en.wikipedia.org/wiki/Free_space


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The Poynting vector appears in the energy-conservation law, orPoynting's theorem,

where J is the current density and u is the electromagnetic energy density.
http://en.wikipedia.org/wiki/Poynting's_theoremhttp://en.wikipedia.org/wiki/Current_densityhttp://www.eeecube.blogspot.com/http://en.wikipedia.org/wiki/Poynting's_theoremhttp://en.wikipedia.org/wiki/Current_density

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