Detailed Notes: Electrostatics | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE) PDF Download

 Page 1


Introduction: 
The electric charge is a fundamental property of matter and charge exist in integral multiple of 
electronic charge. Electrostatics can be defined as the study of electric charges at rest. Electric fields 
have their sources in electric charges. 
(Note: Almost all real electric fields vary to some extent with time. However, for many problems, 
the field variation is slow and the field may be considered as static. For some other cases spatial 
distribution is nearly same as for the static case even though the actual field may vary with time. 
Such cases are termed as quasi-static.) 
In this chapter we first study two fundamental laws governing the electrostatic fields, viz, (1) 
Coulomb's Law and (2) Gauss's Law. Both these law have experimental basis. Coulomb's law is 
applicable in finding electric field due to any charge distribution, Gauss's law is easier to use when 
the distribution is symmetrical 
. 
Coulomb's Law : 
Statement: 
Coulomb's Law states that the force between two point charges Q1and Q2 is directly proportional to 
the product of the charges and inversely proportional to the square of the distance between them. 
Point charge is a hypothetical charge located at a single point in space. It is an idealized model of a 
particle having an electric charge. 
Mathematically, 
 
 
Where k is the proportionality constant. And , is called the permittivity of free space 
In SI units, Q1 and Q2 are expressed in Coulombs(C) and R is in meters. 
Force F is in Newton’s (N) 
(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). 
Page 2


Introduction: 
The electric charge is a fundamental property of matter and charge exist in integral multiple of 
electronic charge. Electrostatics can be defined as the study of electric charges at rest. Electric fields 
have their sources in electric charges. 
(Note: Almost all real electric fields vary to some extent with time. However, for many problems, 
the field variation is slow and the field may be considered as static. For some other cases spatial 
distribution is nearly same as for the static case even though the actual field may vary with time. 
Such cases are termed as quasi-static.) 
In this chapter we first study two fundamental laws governing the electrostatic fields, viz, (1) 
Coulomb's Law and (2) Gauss's Law. Both these law have experimental basis. Coulomb's law is 
applicable in finding electric field due to any charge distribution, Gauss's law is easier to use when 
the distribution is symmetrical 
. 
Coulomb's Law : 
Statement: 
Coulomb's Law states that the force between two point charges Q1and Q2 is directly proportional to 
the product of the charges and inversely proportional to the square of the distance between them. 
Point charge is a hypothetical charge located at a single point in space. It is an idealized model of a 
particle having an electric charge. 
Mathematically, 
 
 
Where k is the proportionality constant. And , is called the permittivity of free space 
In SI units, Q1 and Q2 are expressed in Coulombs(C) and R is in meters. 
Force F is in Newton’s (N) 
(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 ....................... (1) 
 
 
As shown in the Figure 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 1: Coulomb's Law 
 
 
The charges are separated by a distance of . We define the unit vectors as 
 
 
and ..................................(2) 
 
 
can be defined as . 
 
Similarly the force on Q1 due to charge Q2 can be calculated and if represents this force then we can 
write 
Force Due to ‘N ‘no.of point charges: 
When we have a number of point charges, to determine the force on a particular charge due to all other 
charges, we apply principle of superposition. If we have N number of charges Q1,Q2,.........QN located 
respectively at the points represented by the position vectors , ,...... , the force experienced by a 
charge Q located at is given by, 
Page 3


Introduction: 
The electric charge is a fundamental property of matter and charge exist in integral multiple of 
electronic charge. Electrostatics can be defined as the study of electric charges at rest. Electric fields 
have their sources in electric charges. 
(Note: Almost all real electric fields vary to some extent with time. However, for many problems, 
the field variation is slow and the field may be considered as static. For some other cases spatial 
distribution is nearly same as for the static case even though the actual field may vary with time. 
Such cases are termed as quasi-static.) 
In this chapter we first study two fundamental laws governing the electrostatic fields, viz, (1) 
Coulomb's Law and (2) Gauss's Law. Both these law have experimental basis. Coulomb's law is 
applicable in finding electric field due to any charge distribution, Gauss's law is easier to use when 
the distribution is symmetrical 
. 
Coulomb's Law : 
Statement: 
Coulomb's Law states that the force between two point charges Q1and Q2 is directly proportional to 
the product of the charges and inversely proportional to the square of the distance between them. 
Point charge is a hypothetical charge located at a single point in space. It is an idealized model of a 
particle having an electric charge. 
Mathematically, 
 
 
Where k is the proportionality constant. And , is called the permittivity of free space 
In SI units, Q1 and Q2 are expressed in Coulombs(C) and R is in meters. 
Force F is in Newton’s (N) 
(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 ....................... (1) 
 
 
As shown in the Figure 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 1: Coulomb's Law 
 
 
The charges are separated by a distance of . We define the unit vectors as 
 
 
and ..................................(2) 
 
 
can be defined as . 
 
Similarly the force on Q1 due to charge Q2 can be calculated and if represents this force then we can 
write 
Force Due to ‘N ‘no.of point charges: 
When we have a number of point charges, to determine the force on a particular charge due to all other 
charges, we apply principle of superposition. If we have N number of charges Q1,Q2,.........QN located 
respectively at the points represented by the position vectors , ,...... , the force experienced by a 
charge Q located at is given by, 
 
 
.................................(3) 
 
Electric Field intensity: 
The electric field intensity or the electric field strength at a point is defined as the force per unit charge. 
That is 
 
 
or, .......................................(4) 
 
The electric field intensity E at a point r (observation point) due a point charge Q located at (source 
point) is given by: 
 
 
..........................................(5) 
 
For a collection of N point charges Q1 ,Q2 ,.........QN located at , ,...... , the electric field intensity at 
point is obtained as 
 
 
........................................(6) 
 
The expression (6) can be modified suitably to compute the electric filed due to a continuous 
distribution of charges. 
In figure 2 we consider a continuous volume distribution of charge (t) in the region denoted as the 
source region. 
 
For an elementary charge , i.e. considering this charge as point charge, we can write the 
field expression as: 
 
 
.............(7) 
Page 4


Introduction: 
The electric charge is a fundamental property of matter and charge exist in integral multiple of 
electronic charge. Electrostatics can be defined as the study of electric charges at rest. Electric fields 
have their sources in electric charges. 
(Note: Almost all real electric fields vary to some extent with time. However, for many problems, 
the field variation is slow and the field may be considered as static. For some other cases spatial 
distribution is nearly same as for the static case even though the actual field may vary with time. 
Such cases are termed as quasi-static.) 
In this chapter we first study two fundamental laws governing the electrostatic fields, viz, (1) 
Coulomb's Law and (2) Gauss's Law. Both these law have experimental basis. Coulomb's law is 
applicable in finding electric field due to any charge distribution, Gauss's law is easier to use when 
the distribution is symmetrical 
. 
Coulomb's Law : 
Statement: 
Coulomb's Law states that the force between two point charges Q1and Q2 is directly proportional to 
the product of the charges and inversely proportional to the square of the distance between them. 
Point charge is a hypothetical charge located at a single point in space. It is an idealized model of a 
particle having an electric charge. 
Mathematically, 
 
 
Where k is the proportionality constant. And , is called the permittivity of free space 
In SI units, Q1 and Q2 are expressed in Coulombs(C) and R is in meters. 
Force F is in Newton’s (N) 
(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 ....................... (1) 
 
 
As shown in the Figure 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 1: Coulomb's Law 
 
 
The charges are separated by a distance of . We define the unit vectors as 
 
 
and ..................................(2) 
 
 
can be defined as . 
 
Similarly the force on Q1 due to charge Q2 can be calculated and if represents this force then we can 
write 
Force Due to ‘N ‘no.of point charges: 
When we have a number of point charges, to determine the force on a particular charge due to all other 
charges, we apply principle of superposition. If we have N number of charges Q1,Q2,.........QN located 
respectively at the points represented by the position vectors , ,...... , the force experienced by a 
charge Q located at is given by, 
 
 
.................................(3) 
 
Electric Field intensity: 
The electric field intensity or the electric field strength at a point is defined as the force per unit charge. 
That is 
 
 
or, .......................................(4) 
 
The electric field intensity E at a point r (observation point) due a point charge Q located at (source 
point) is given by: 
 
 
..........................................(5) 
 
For a collection of N point charges Q1 ,Q2 ,.........QN located at , ,...... , the electric field intensity at 
point is obtained as 
 
 
........................................(6) 
 
The expression (6) can be modified suitably to compute the electric filed due to a continuous 
distribution of charges. 
In figure 2 we consider a continuous volume distribution of charge (t) in the region denoted as the 
source region. 
 
For an elementary charge , i.e. considering this charge as point charge, we can write the 
field expression as: 
 
 
.............(7) 
 
 
 
Fig 2: Continuous Volume Distribution of Charge 
 
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: 
 
 
..........................................(8) 
 
Similar technique can be adopted when the charge distribution is in the form of a line charge density or a 
surface charge density. 
 
 
........................................(9) 
 
 
 
........................................(10) 
 
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: 
................................................(11) 
 
We define the electric flux as 
 
 
.....................................(12) 
Page 5


Introduction: 
The electric charge is a fundamental property of matter and charge exist in integral multiple of 
electronic charge. Electrostatics can be defined as the study of electric charges at rest. Electric fields 
have their sources in electric charges. 
(Note: Almost all real electric fields vary to some extent with time. However, for many problems, 
the field variation is slow and the field may be considered as static. For some other cases spatial 
distribution is nearly same as for the static case even though the actual field may vary with time. 
Such cases are termed as quasi-static.) 
In this chapter we first study two fundamental laws governing the electrostatic fields, viz, (1) 
Coulomb's Law and (2) Gauss's Law. Both these law have experimental basis. Coulomb's law is 
applicable in finding electric field due to any charge distribution, Gauss's law is easier to use when 
the distribution is symmetrical 
. 
Coulomb's Law : 
Statement: 
Coulomb's Law states that the force between two point charges Q1and Q2 is directly proportional to 
the product of the charges and inversely proportional to the square of the distance between them. 
Point charge is a hypothetical charge located at a single point in space. It is an idealized model of a 
particle having an electric charge. 
Mathematically, 
 
 
Where k is the proportionality constant. And , is called the permittivity of free space 
In SI units, Q1 and Q2 are expressed in Coulombs(C) and R is in meters. 
Force F is in Newton’s (N) 
(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 ....................... (1) 
 
 
As shown in the Figure 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 1: Coulomb's Law 
 
 
The charges are separated by a distance of . We define the unit vectors as 
 
 
and ..................................(2) 
 
 
can be defined as . 
 
Similarly the force on Q1 due to charge Q2 can be calculated and if represents this force then we can 
write 
Force Due to ‘N ‘no.of point charges: 
When we have a number of point charges, to determine the force on a particular charge due to all other 
charges, we apply principle of superposition. If we have N number of charges Q1,Q2,.........QN located 
respectively at the points represented by the position vectors , ,...... , the force experienced by a 
charge Q located at is given by, 
 
 
.................................(3) 
 
Electric Field intensity: 
The electric field intensity or the electric field strength at a point is defined as the force per unit charge. 
That is 
 
 
or, .......................................(4) 
 
The electric field intensity E at a point r (observation point) due a point charge Q located at (source 
point) is given by: 
 
 
..........................................(5) 
 
For a collection of N point charges Q1 ,Q2 ,.........QN located at , ,...... , the electric field intensity at 
point is obtained as 
 
 
........................................(6) 
 
The expression (6) can be modified suitably to compute the electric filed due to a continuous 
distribution of charges. 
In figure 2 we consider a continuous volume distribution of charge (t) in the region denoted as the 
source region. 
 
For an elementary charge , i.e. considering this charge as point charge, we can write the 
field expression as: 
 
 
.............(7) 
 
 
 
Fig 2: Continuous Volume Distribution of Charge 
 
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: 
 
 
..........................................(8) 
 
Similar technique can be adopted when the charge distribution is in the form of a line charge density or a 
surface charge density. 
 
 
........................................(9) 
 
 
 
........................................(10) 
 
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: 
................................................(11) 
 
We define the electric flux as 
 
 
.....................................(12) 
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 3: Gauss's Law 
Let us consider a point charge Q located in an isotropic homogeneous medium of dielectric constant . 
The flux density at a distance r on a surface enclosing the charge is given by 
 
...............................................(13) 
 
If we consider an elementary area ds, the amount of flux passing through the elementary area is given by 
 
 
.....................................(14) 
 
 
But ,  is  the  elementary  solid  angle  subtended  by  the  area at the location of Q. 
Therefore we can write 
For a closed surface enclosing the charge, we can write 
 
Which can seen to be same as what we have stated in the definition of Gauss's Law. 
 
Application of Gauss's Law: 
 
Gauss's  law  is  particularly  useful   in  computing or where the charge distribution has some