Electromagnetics Chapter 1

ElectromagneticsChapter 1Introduction

1.3 The Nature of Electromagnetism (EM)There are four fundamental forces in nature :1- The nuclear force : is the strongest of the four andlimited to submicroscopic systems(nuclei).2- The electromagnetic force : its strength is 10 2 ofthe nuclear force and is found in microscopicsystems, such as atoms and molecules .3- The weak-interaction force : its strength is 10 14of the nuclear force and plays a role in interactionsinvolving radioactive particles.

4- The gravitational force : is the weakest of the fourforces (about 10 41 of the nuclear force), but it is thedominant force in macroscopic system ,such as the solarsystem.*** our interest in this course is with electromagneticforce which consists of electrical force 𝐹𝑒 and magneticforce 𝐹𝑚 .,butWe first look at the gravitational force properties becausethey provide a useful analogue to those of the electromagneticforce .

The Gravitational Force: A Useful AnalogueAccording to Newton’s law of gravity, the gravitational force𝐹𝑔21 acting on mass 𝑚2 due to mass 𝑚1 at distance 𝑅12from 𝑚2 is given by :

𝑭𝒈𝟐𝟏 𝑮𝒎𝟏 𝒎𝟐 𝑹𝟏𝟐(𝑹𝟏𝟐 )𝟐(N),whereG : is the universal gravitational constant.G 𝟔. 𝟔𝟕 𝟏𝟎 𝟏𝟏𝑵. 𝒎𝟐 /𝒌𝒈𝟐𝑹𝟏𝟐 : is a unit vector that points from 𝑚1to 𝑚2 .

*** The minus sign indicates that thegravitational force is attractive.Source offorce,so𝑭𝒈𝟐𝟏 - 𝑭𝒈𝟏𝟐Massexperiencingthe force,where 𝑭𝒈𝟏𝟐 is the force acting on mass 𝑚1 due togravitational pull of mass 𝑚2 .

The gravitational force does not require anycontact between the interacting particles. Thisis called action-at-a-distance. This phenomenon of direct action at a distanceled to a concept called fields.

Figure 1-3 shows a gravitational field 𝜓1 induced by a mass𝑚1 that does not physically emanate(release) from the object,but its influence exists at every point in space.

At a distance R from 𝑚1 , the field 𝜓1 is avector defined as :𝑮𝒎𝟏𝝍𝟏 𝑹 𝟐𝑹(𝑵 𝒌𝒈)𝑹 is a unit vector which points in the radialdirection from 𝑚1 and therefore -𝑹 pointstoward 𝑚1 .,where

The force due to 𝝍𝟏 acting on a mass 𝑚2 at adistance 𝑅 𝑅12 along the direction 𝑅 𝑅12is:𝑭𝒈𝟐𝟏 𝝍𝟏 𝒎𝟐 𝑮𝒎𝟏 𝒎𝟐 𝑹𝟏𝟐(𝑹𝟏𝟐 )𝟐(N)

At any point in space, the force 𝑭𝒈 acting on amass m placed at that point is related to 𝝍 by:𝝍 𝑭𝒈𝒎*** The force 𝑭𝒈 is due to a single mass or adistribution of many masses.

Electric FieldsUntil now, we have learned that mass is the sourceof the gravitational field ,but what is about theelectric field?! ,what is its source?!what is thedifference between its source and the gravitationalfield source?! Is it a big difference?!

The source of electrical field is electric charge. It is as gravitational field varies inversely withthe square of the distance from its respectivesource. Electric charge may have positive or negativepolarity whereas mass does not exhibit such aproperty.

According to atomic physics, all matter contains of neutrons,positively charged protons and negatively charged electrons. Fundamental quantity of charge is that of a single electron denotedby the letter 𝒆. The unit of electric charge is coulomb (C),named in the honor of18th century French scientist Charles Augustin de Coulomb. The magnitude of 𝒆 is :𝒆 𝟏. 𝟔 𝟏𝟎 𝟏𝟗𝑪 .

Electron charge: 𝑞𝑒 𝑒 Proton charge: 𝑞p 𝑒

Coulomb’s Law

These properties constitute Coulomb’s lawwhich can be expressed mathematically by thefollowing equation:

𝑭𝒆𝟏𝟐 - 𝑭𝒆𝟐𝟏,where𝐹𝑒21 :𝑅12 :𝑅12 𝜀0 :is the electrical force acting on charge 𝑞2 due to charge 𝑞1 .is the distance between the two charges .is a unit vector pointing from charge 𝑞1 to charge 𝑞2 .is the electrical permittivity of free space.(𝜺𝟎 𝟖. 𝟖𝟒𝟓 𝟏𝟎 𝟏𝟐𝐟𝐚𝐫𝐚𝐝 𝐩𝐞𝐫 𝐦𝐞𝐭𝐞𝐫 𝑭 𝒎 )

In analogy to gravitational field ψ, theelectric field intensity 𝑬 due to any charge qcan be expressed as:𝒒𝑬 𝑹𝟒𝝅𝜺𝒐 𝑹𝟐𝑽 𝒎Where,𝑹 : is the distance the chargeand observation point.𝑹 : is the radial unit vectorpointing away fromthe charge.(𝐢𝐧 𝐟𝐫𝐞𝐞 𝐬𝐩𝐚𝐜𝐞)

Electric charge exhibits two importantproperties:1- the low of conservation of electric chargewhich states that the net electric charge canneither be created nor destroyed.if a volume contains 𝑛𝑝 protons and 𝑛𝑝 electrons, then the totalcharge is :𝒒 𝒏𝒑 𝒆 𝒏𝒆 𝒆 𝒏𝒑 𝒏𝒆 𝒆𝑪 .Note that .Even if some of the protons were to combine with an equalnumber of electrons to produce neutrons or other particles, thenet charge 𝒒 remains unchanged.

2- The principle of linear superposition, whichstates that the total vector electric field at a pointin space due to a system of point charges is equalto the vector sum of the electric fields at thatpoint due to the individual charges.We will use this property in future chapters to compute theelectric field due to complex distributions of charge.

Until now, we have described the electric fieldinduced by an electric charge when in freespace. What happens if a positive point charge isplaced in a material composed of atoms?!

Before adding a test charge ( positive charge)inside the material, the material is electricallyneutral. When a test charge is placed, the atoms aredistorted and polarized as shown in figure( 1-6).

The degree of polarization depends on the distancebetween the atom and the isolated point charge. Polarization is the process of electric dipoleformation. Dipoles of the atoms or ( molecules ) counteract thefield due to the point charge. Therefore, the electric field inside the material isdifferent from that of free space.

To extend our equation for electric field fromfree space to inside the material, we replace𝜀0 𝑏𝑦 𝜀

𝜀 is defined as :𝜺 𝜺𝒓 𝜺 𝟎(𝑭 𝒎),where 𝜺𝒓 is a dimensionless quantity called the“relative permittivity” or “dielectric constant” ofthe material.*** For vacuum, 𝜺𝒓 1*** For air near earth’s surface, 𝜺𝒓 1.0006

Electric Flux Density: In addition to the electric field intensity E, wewill often find it convenient to also use arelated quantity called the electric flux densityD, given by :𝑫 𝜺𝑬(𝑪 𝒎𝟐 )Note that .E and D constitute one of two fundamental pairs ofelectromagnetic field quantities.

Magnetic field As early as 800 B.C.,first magnetic stones werediscovered by Greeks. We call these stonesnow “magnetite” (𝐹𝑒3 𝑂4 ). Each magnet has two poles:north and south poles.

It is impossible to separate N and S. If themagnet is broken into pieces, each piece hasits own north and south poles. The magnetic lines encircling a magnet are calledmagnetic field lines.

These lines represent the existence of amagnetic field called the magnetic fluxdensity B. A magnetic field not only exists aroundpermanent magnets but can also be created byelectric current. This connection between electricity andmagnetism was first discovered by Danishscientist Hans Oersted in 1819.

Shortly after oersted,French scientists JeanBaptiste Biot and Felix Savart developed anexpression relating 𝐼 and 𝐵. This expression is now called “Biot-Savartlaw”:

The unit of B is tesla (T). 𝜇0 : is the magnetic permeability of free space.𝝁𝟎 𝟒𝝅 𝟏𝟎 𝟕 𝑟: is the radial distance from the current. 𝜙 is an azimuthal unit vector denoting the fact thatthe magnetic field direction is tangential to thecircle surrounding the current.

Electric and magnetic fields are connectedthrough the speed of light:

The magnetic permeability 𝜇 accounts for magnetizationproperties of a material.*** A Nonmagnetic material has 𝜇 𝜇0 .*** A Ferromagnetic material like iron has 𝜇 𝜇0 . In analogy to permittivity of material 𝜺, 𝜇 can bedefined as :𝝁 𝝁𝒓 𝝁𝟎(𝑯 𝒎)Where, 𝜇𝑟 is dimensionless quantity called relativemagnetic permeability of the material.

We have learned before that E and D constituteone of two pairs of electromagnetic fieldquantities. The second pair is B and the magnetic fieldintensity H,which are related to each otherthrough 𝜇:𝑩 𝝁𝑯(𝑻)

Static and Dynamic Fields There are three branches of electromagnetics:1- Electrostatics2- Magnetostatics3- Dynamics (Time-varying fields).

The electric field E is governed by charge q.So if the charge doesn’t change with time, Eremains constant. This corresponds to “Electrostatics”: 𝑞 0 𝐸 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡

Also,we have learned that the magnetic field 𝑯is governed by the current 𝐼. So if 𝐼 is constantB is constant. This corresponds to “Magnetostatics”: 𝑰 𝟎 𝑩 𝒊𝒔 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝒕

Now we consider the case when 𝐼 changeswith time.𝒅𝒒𝒅𝒕Since 𝑰 , it means that amount of chargepresent in a given section of a wire ,varies alsowith time. So E varies with time .In general, A time-varying electric field will generate atime-varying magnetic field and vice versa.

Summary of the three branches of electromagnetics

Note that .Under static conditions, induced electric and magnetic fields areindependent;under Dynamic conditions, fields become coupled.Also,

Conductivity 𝝈:The conductivity indicates how easy charges movein material and it is measured in Siemens permeter(S/m).** If 𝝈 𝟎 , charges do not move more than atomicdistances and the material is called "perfectdielectric”.** If 𝝈 ,the charges can move freely inside thematerial which is then called a perfect conductor.

𝜀, 𝜇 and 𝜎 are the “constitutive parameters” of a material.A medium is said to be homogeneous if its constitutiveparameters are constant throughout the medium.

Electromagnetics Chapter 1 Introduction . 1.3 The Nature of Electromagnetism (EM) . There are three branches of electromagnetics: 1- Electrostatics 2- Magnetostatics 3- Dynamics (Time-varying fields). The electric field E is governed by charge q.