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PROVIDED BY BRAINZORP

ELECTROMAGNETIC FIELDS THEORY
UNIT - 1 ELECTROSTATICS 2

EE1005 ELECTROMAGNETIC FIELDS THEORY
CONTENTS Vector Calculus Coordinate Systems Differential elements of the coordinate systems Electric field intensity Coulomb’s law Electric flux density Gauss’s law and its applications Divergence and divergence theorem Potential and potential gradient Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 3

Syllabus UNIT –1 - “ ELECTROSTATICS” Introduction to various co ordinate systems – Coulomb’s law – Electric field intensity – Electric fields due to point, line , surface and volume charge distributions – Electric flux density – Gauss’s law and its applications – Electric potential – Potential gradient – Divergence – divergence theorem. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 4

Scalar Quantity A quantity that has only magnitude but no direction. Example: time, mass, distance, temperature and population are scalars. Scalar is represented by a letter. e.g., A, B Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 5

Vector Quantity A quantity that has both magnitude and direction. Example: Velocity, force, displacement and electric field intensity. Vector is represent by a letter such as A, B, or It can also be written as where A is which is the magnitude and is unit vector Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 6

Unit Vector A unit vector along A is defined as a vector whose magnitude is unity (i.e., 1) and its direction is along A. It can be written as a A or Thus, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 7

Vector Addition The sum of two vectors for example vectors A and B can be obtain by moving one of them so that its terminal point (tip) coincides with the initial point (tail) of the other Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 8

Vector Subtraction Vector subtraction is similarly carried out as D = A – B = A + (-B) Figure ( C ) shows that vector D is a vector that must be added to B to give Vector A. So if vector A and B are placed tail to tail then vector D is a vector that runs from the tip of B to A. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 9

Vector Multiplication
Scalar (dot ) product (A•B) Vector (cross) product (A X B) Scalar triple product A • (B X C) Vector triple product A X (B X C) Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 10

Multiplication of a vector by a scalar
Multiplication of a scalar k to a vector A gives a vector that points in the same direction as A and magnitude equal to |kA|. The division of a vector by a scalar quantity is a multiplication of the vector by the reciprocal of the scalar quantity. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 11

Scalar Product The dot product of two vectors and , written as is defined as the product of the magnitude of and , and the projection of onto (or vice versa). Thus, Where, θ is the angle between and . The result of dot product is a scalar quantity. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 12

Vector Product The cross (or vector) product of two vectors A and B, written as is defined as where; a unit vector perpendicular to the plane that contains the two vectors. The direction of is taken as the direction of the right thumb (using right-hand rule) The product of cross product is a vector Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 13

Right Hand Rule Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 14

Components of a Vector A direct application of vector product is in determining the projection (or component) of a vector in a given direction. The projection can be scalar or vector. Given a vector A, we define the scalar component AB of A along vector B as, AB = A cos θAB = |A||aB| cos θAB or AB = A·aB Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 15

Dot product If and then which is obtained by multiplying A and B component by component. It follows that modulus of a vector is Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 16

Cross Product If A=(Ax, Ay, Az), B=(Bx, By, Bz) then Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 17

Cross Product Cross product of the unit vectors yield: Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 18

Example 1 Given three vectors P = Q = R= Determine a) (P+Q) X (P-Q) b) Q•(R X P) c) P•(Q X R) d) e) P X( Q X R) f) A unit vector perpendicular to both Q and R Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 19

Solution Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 20

Solution (Continued) Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 21

Solution (Continued) To find the determinant of a 3 X 3 matrix, we repeat the first two rows and cross multiply; when the cross multiplication is from right to left, the result should be negated as shown below. This technique of finding a determinant applies only to a 3 X 3 matrix. Hence, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 22

Solution (Continued) (c) or Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 23

Solution (Continued) EE1005 ELECTROMAGNETIC FIELDS THEORY Chapter 1 24

Solution (Continued) Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 25

Need for coordinate systems
To describe a vector accurately and to express a vector in terms of its components, it is necessary to have some reference directions. Such directions are represented in terms of coordinate systems. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 26

Types of coordinate systems
The various coordinate systems used often in electromagnetic fields are, i) Cartesian coordinate system, ii) Cylindrical coordinate system, iii) Spherical coordinate system Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 27

Cartesian coordinates
Very convenient when dealing with the problems having rectangular symmetry. This system has three coordinate axes represented by (x,y,z) which are mutually at right angles to each other. These three axes intersect at a common point called origin of the system. There are two types of such system called, i) Right handed system , ii) Left handed system. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 28

Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 29

Cartesian coordinate In this system, x = 0 indicates 2 dimensional y - z plane, y = 0 indicates 2 dimensional x - z plane and z = 0 indicates 2 dimensional x - y plane. A vector in Cartesian coordinates can be written as (Ax, Ay, Az) or Axax + Ayay + Azaz. The ranges of (x,y,z) are as follows: -∞ < x < ∞ -∞ < y < ∞ -∞ < z < ∞ Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 31

Cylindrical Coordinates
Very convenient when dealing with problems having cylindrical symmetry. A point P in cylindrical coordinates is represented as (r, Φ, z) where r: is the radius of the cylinder; radial displacement from the z-axis Φ: azimuthal angle or the angular displacement from x-axis z : vertical displacement z from the origin (as in the Cartesian system). Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 32

Cylindrical coordinates

The range of the variables are 0 ≤ r < ∞, 0 ≤ Φ < 2π , -∞ < z < ∞ vector in cylindrical coordinates can be written as (Ar, Aφ, Az) or Arar + Aφaφ+ Azaz The magnitude of is Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 35

Relationships Between Variables
The relationships between the variables (x,y,z) of the Cartesian coordinate system and the cylindrical system (ρ, φ , z) are obtained as So a point P (3, 4, 5) in Cartesian coordinate is the same as? Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 38

Relationships Between Variables
So a point P (3, 4, 5) in Cartesian coordinate is the same as P ( 5, 0.927,5) in cylindrical coordinate. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 39

Spherical Coordinates
The spherical coordinate system is used dealing with problems having a degree of spherical symmetry. Point P represented as (r,θ,φ) where r : the distance from the origin, θ : called the colatitude is the angle between z-axis and vector of P, Φ : azimuthal angle or the angular displacement from x-axis (the same azimuthal angle in cylindrical coordinates). Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 40

Relation between cartesian coordinate system: Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 43

Point transformation Point transformation between cylinder and spherical coordinate is given by Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 44

Differential Elements
In vector calculus the differential elements in length, area and volume are useful. They are defined in the Cartesian, cylindrical and spherical coordinate Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 45

Cartesian Coordinates
az ay ax Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 46

Chapter 1 EE 1005 ELECTROMAGNETIC FIELDS THEORY 48

P Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 49

Chapter 1 BEE 3113ELECTROMAGNETIC FIELDS THEORY 52

Electrostatics Electrostatics is the branch of electromagnetics dealing with the effects of electric charges at rest. The fundamental law of electrostatics is Coulomb’s law. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 55

Electric Charge Electrical phenomena caused by friction are part of our everyday lives, and can be understood in terms of electrical charge. The effects of electrical charge can be observed in the attraction/repulsion of various objects when “charged.” Charge comes in two varieties called “positive” and “negative.” Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 56

Electric Charge Objects carrying a net positive charge attract those carrying a net negative charge and repel those carrying a net positive charge. Objects carrying a net negative charge attract those carrying a net positive charge and repel those carrying a net negative charge. On an atomic scale, electrons are negatively charged and nuclei are positively charged. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 57

Electric Charge Electric charge is the physical property of the matter that causes it to experience a force when placed in an electromagnetic field. e =  C. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 58

Coulomb’s Law Coulomb’s law is the “law of action” between charged bodies. The coulombs law states that the force between the two point charges Q1 and Q2 , Acts along the line joining the two points, Is directly proportional to the product of the two charges, Is inversely proportional to the square of the distance between them. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 59

Coulomb’s law Point Charge: A point charge is a electric charge that occupies a region of space which is negligibly small compared to the distance between the point charge and any other object. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 60

BEE 3113ELECTROMAGNETIC FIELDS THEORY
Coulomb’s law Consider the two point charges Q1 and Q2, Mathematically the force between the charges can be explained as, Where, Q1Q2 - the product of the charges, Distance between the two charges. R Chapter 1 BEE 3113ELECTROMAGNETIC FIELDS THEORY 61

Coulomb’s law It also states that this force depends on the medium in which the point charges are located. where, K – Constant of proportionality Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 62

Coulomb’s law where, ε - Permittivity of the medium in which charges are located in Farad/metre (absolute permittivity) ε0 – Permittivity of free space or vacuum εr - Relative permittivity or dielectric constant of the medium Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 63

Vector form Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 64

Coulomb’s law The force on Q1 due to Q2 is equal in magnitude but opposite in direction to the force on Q2 due to Q1. 1 2 F - = Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 65

Electric Field Intensity
Electric field intensity is defined as the strength of an electric field at any point. It is equal to the force per unit charge experienced by a test charge placed at that point. The unit of measurement is the volt/meter. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 66

Charge distributions Point Charge:- If the dimensions of a surface carrying charge is very very small when compare to the surface surrounding it, then it is treated as a point. Such a charge is called as point charge. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 67

Charge distributions Line Charge:- If the charge may be spreaded all along a line which may be finite or infinite, then such a charge is called as line charge. Line Charges Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 68

Charge distributions Surface charge:- If the charge is uniformly distributed along a two dimensional surface then it is called as surface charge or a sheet of charge. Surface charges Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 69

Charge distributions Volume Charge:- If the charge is distributed uniformly over a volume, it is called volume charge. Volume charge Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 70

E due to line charge Line charge distribution Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 71

E due to line charge Differential charge: Differential field strength: Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 72

E due to surface charge Surface Charge Distribution Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 73

E due to surface charge Differential charge: Differential field strength: Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 74

E due to volume charge Volume Charge Distribution Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 75

E due to volume charge Differential charge: Differential field strength: Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 76

E due to infinite line charge

Consider an infinitely straight line carrying uniform line charge having density ρL C/m. Let this line lies along z axis from -∞ to ∞ and hence called infinite line charge. Let point P is on Y axis at which electric field intensity is to be determined. The distance of point P from the origin is ‘r’. Consider a small differential length dl carrying a charge dQ along the line. It is along Z axis hence, dl = dz. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 78

Symmetry of charges in z direction Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 79

Therefore, dQ = ρL dl = ρL dz. The coordinates of dQ are (0,0,z) while the coordinates of point P are (0,r,0). Hence the distance vector R can be written as, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 80

Therefore, we get the differential field intensity, The field intensity of the infinite line charge is given by, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 81

For the integration, use the following substitution, Here, r is not the variable of integration. By changing the limits, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 82

Therefore, Hence the result can be expressed as, Chapter 1 BEE 3113ELECTROMAGNETIC FIELDS THEORY 84

E due to charged circular ring

Consider a charged circular ring of radius r placed in XY plane with Centre as origin, carrying a charge uniformly along its circumference. The charge density is ρL C/m. The point P is at a perpendicular distance z from the ring as shown in the figure. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 86

Consider a small differential length dl on this ring. The charge on it is dQ. where, R – Distance of point P from dl. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 87

Consider the cylindrical coordinate system. For dl, we are moving in ϕ direction, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 88

Here, R can be obtained from it two components, in cylindrical system. They are, i) Distance r in the direction of - ar, radially inward. i.e., -r. ar ii) Distance z in the direction of az vector. i.e., z az. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 89

Therefore, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 90

By substituting the values, we get, Now, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 91

Therefore, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 92

Therefore, where, r – Radius of the ring, z – Perpendicular distance of point P from the ring. This is the electric field at a point P (0,0,z) due to the circular ring of radius r placed in XY plane. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 93

Electric Field due to infinite sheet of charge
Consider an infinite sheet of charge having uniform charge density ρS in Coulomb per metre square. It is placed in XY plane. Let us use cylindrical coordinates. The point P at which E to be calculated is on z axis. Consider the differential surface area ds carrying a charge dQ. The normal direction to dS is z direction hence dS normal to z direction is r dr dϕ . Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 94

The distance vector R has two components. i) The radial component r along – ar . i.e., -r ar . ii)The component z along az. i.e., z az . Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 96

With these two components R can be obtained from the differential area towards point P as, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 97

The dE can be written as, For a infinite sheet in XY plane, r varies from 0 to ∞ while ϕ varies from 0 to 2π. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 98

As there is symmetry about z axis from all radial direction, all ar components of E are going to cancel each other and the net E will not have any radial component. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 99

Therefore the E can be written as, By changing the limits, we get, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 100

Therefore on integration, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 101

It can be written as, where, an – direction normal to the surface charge For the points below XY plane, an = - az Therefore, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 102

EE 1005 ELECTROMAGNETIC FIELDS THEORY
Electric Flux The total number of lines of force in any particular electric field is called electric flux. It is represented by the symbol ψ. The unit of flux is Coulomb. Electric flux lines Chapter 1 EE 1005 ELECTROMAGNETIC FIELDS THEORY 103

Properties of electric flux
The flux lines start from positive charge and terminate at negative charge. If the negative charge is absent, then the flux lines terminate at infinity. There are more number of lines. i.e., crowding of lines if the electric field is stronger. These lines are parallel and never cross each other. The lines are independent of the medium in which the charges are placed. x Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 104 104

Electric Flux Density Electric flux density or displacement density is defined as the electric flux per unit area. D = ψ/S Coulomb / metre square. where, Ψ – Total Flux, S – Surface Area For a sphere surface area, A = 4 Π r2 D = Q / 4 Π r2 But, E = Q / 4 Π ε0 r2 D = ε0 E Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 105

Electric flux Density In vector form, where, dψ – total flux lines crossing normal to the surface area dS. dS – differential surface area an – unit vector direction normal to the differential surface area. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 106

D due to a line charge Consider a line charge having density ρL C / m. Total charge along the line is given by, If the line charge is infinite, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 107

D due to surface charge Consider a surface charge having density of ρS C / m 2. Total charge along the surface is given by, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 108

D due to surface charge If the sheet of charge is infinite then, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 109

D due to volume charge Consider a charge enclosed by a volume, with a uniform charge density ρv C / m 3 Then the total charge enclosed by the volume is given by, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 110

Divergence Divergence is the outflow of flux from a small closed surface area (per unit volume) as volume shrinks to zero. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 111

Divergence Example for divergence:- Water leaving a bathtub (incompressible) Air leaving a punctured tire (positive divergence) Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 112

BEE 3113ELECTROMAGNETIC FIELDS THEORY
Divergence Mathematical expression for divergence, Surface integral as the volume element approaches zero D is the vector flux density Chapter 1 BEE 3113ELECTROMAGNETIC FIELDS THEORY 113

Divergence For Cartesian coordinates, For cylindrical coordinates, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 114

Divergence For spherical coordinates, Divergence is a scalar. Dot product of two vectors will give the resultant as a scalar as divergence. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 115

Del Operator Written as is the vector differential operator. Also known as the gradient operator. The operator in useful in defining: The gradient of the scalar V, written as V The divergence of a vector A, written as The curl of a vector A, written as The laplacian of a scalar V, written as writtenas Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 116 116

Divergence Theorem The volume integral of the divergence of a vector field over a volume is equal to the surface integral of the normal component of this vector over the surface bounding the volume. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 117

Divergence theorem From the gauss law we can write, Q = ∫∫ D. dS (1) While the charge enclosed in a volume is given by, Q = ∫∫∫ ρv dv (2) But according to Gauss’s law in the point form, D = ρv (3) Using equation (2), Q = ∫∫∫ ( D) dv (4) Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 118

Divergence theorem Equating equations (1) and (4), ∫∫ D. dS = ∫∫∫ ( D) dv The above equation is called divergence theorem. It is also called as Gauss - Ostrogradsky theorem. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 119

Gauss’s Law “The electric flux passing through any closed surface is equal to the total charge enclosed by that surface.” Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 120

Gauss’s Law Closed Surface Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 121

Gauss’s Law Consider a small element of area ds in a plane surface having charge Q and P be a point in the element. At every point on the surface the electric flux density D will have value Ds. Let Ds make an angle θ with ds. The flux crossing ds is the product of the normal component of Ds and ds. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 122

Gauss’s law Therefore, dψ = Ds normal .ds = Ds cosθ ds dψ = Ds. ds …. (dot product) The flux passing through the closed surface is given by, ψ = ∫ dψ = ∫∫ Ds . ds ψ = Q In terms of volume charge density, ψ = ∫∫∫ ρv. dv = Q Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 123

Point form of Gauss’s Law
The divergence of electric flux density in a medium at a point (differential volume shrinking to zero), is equal to the volume charge density (charge per unit volume) at the same point. This point form is also called as differential form. Therefore, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 124

Applications of Gauss’s Law
Point Charge:- Let a point charge Q be located at the origin. To determine D and to apply Gauss’s law, consider a spherical surface around Q with Centre as origin. This spherical surface is Gaussian surface and it satisfies the required condition. The D is always directed radially outwards along ar which is normal to the spherical surface at any point P on the surface. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 125

Point Charge Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 126

Consider a differential surface area ds. The direction normal to the surface ds is ar (Spherical coordinate system). The radius of the sphere is (r = a). Then, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 127

The D due to a point charge is given by, Note that, θ’ is the angle between D and ds. Where, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 128

The normal to ds is ar, while D also acting along ar, hence angle between D and ds, i.e., 0 degree. Therefore, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 129

Therefore, This proves the Gauss’s law. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 130

Infinite line of charge:- Consider an infinite line charge of density ρl C/m lying along z axis from - ∞ to ∞. Consider the Gaussian surface as a right circular cylinder with z – axis as its axis and radius r. The length of the cylinder is L. The flux density on any point on the surface is directed radially outwards. i.e., in the ar direction according to the cylindrical coordinate system. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 131

Infinite line charge Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 132

Consider differential surface area ds which is at a radial distance r from the line charge. The direction normal to dS is ar. As the line charge is along z axis, there cannot be any component of D in z direction. So D has only radial component. Now, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 133

The integration is to be evaluated for side surface, top surface and bottom surface. Therefore, Now, Now, Dr is constant over the side surface. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 134

As D has only radial component and no component along az and – az, hence integrations over top and bottom surfaces is zero. Therefore, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 135

Therefore, we get, The results are same as obtained from the Coulomb’s law. .. Due to infinite charge Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 136

Co-axial Cable:- Consider the two co-axial cylindrical conductors forming a co-axial cable. The radius of the inner conductor is ‘a’ while the radius of the outer conductor is ‘b’. The length of the cable is L. The charge distribution on the outer surface of the inner conductor is having density ρs C/m. The total outer surface area of the inner conductor is 2Πa L. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 137

Co-axial cables Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 138

Hence,ρs can be expressed in terms of ρL. Therefore, Thus the line charge density of inner conductor is ρL C/m. Consider the right circular cylinder of length L as the Gaussian surface. Due to the symmetry, D has only radial component. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 139

Application of Gauss’s Law
Now, we can write, The total charge on the inner conductor is to be obtained by evaluating the surface integral of the surface charge distribution. Now, …… (a) Where, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 140

Therefore, By equating (a) and (b), …….. (b) Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 141

But, This is same as obtained for infinite line charge. Every flux line starting from the positive charge on the inner cylinder must terminate on the negative charge on the inner surface of the outer cylinder. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 142

Hence the total charge on the inner surface of the outer cylinder is, But, Therefore, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 143

Infinite Sheet of Charge:- Consider the infinite sheet of charge density ρs C/m square, lying in the z = 0 plane. i.e., XY plane. Consider a rectangular box as a Gaussian surface which is cut by the sheet of charge to give ds = dx dy. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 144

Infinite sheet of charge Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 145

But, The results are same as obtained by the coulomb’s law for infinite sheet of charge. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 147

Differential Volume element:- Differential Volume element Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 148

According to Gauss’s Law, The total surface integral is to be evaluated over six surfaces front, back, left, right , top and bottom. Therefore, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 149

Consider the front surface of the differential element. Though D is varying with distance, for small surface like front surface it can be assumed constant. Now, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 150

Applications of Gauss’s law
It has been mentioned that Dx , front is changing in X direction. At P, it is Dx0 while on the front surface it will change and given by, Therefore, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 151

Consider the integral over the back surface, Therefore, where, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 152

Now, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 153

Now, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 154

But, The charge enclosed in a volume is given by, This result leads to the concept of divergence. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 155

Electric Potential The electric potential may be illustrated with the help of the following concepts. Work done Potential Difference Absolute potential Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 156

Work done The work is said to be done when the test charge is moved against the electric field. The potential energy of the test charge is nothing but the work done in moving the charge against the electric field. i.e., work done in moving the test charge against the direction of electric field over the distance from initial position to final position. It is measured in joules. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 157

Differential Work done
If a charge Q is moved from initial position to final position, against the direction of electric field E then the total work done is obtained by integrating the differential work done over a distance from initial position to the final position. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 158

Line Integral Consider that the charge is moved from initial position B to the final position A, against the electric field E the work done is given by, This is called line integral. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 159

Potential Difference The work done in moving unit charge from point B to A in the field of E is called Potential difference between the points B and A. It is denoted by V and its unit is J / C or V. If B is the initial point and A is the final point then the P.D is denoted by VAB. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 160

Potential due to Point charge
Consider a point charge, located at the origin of a spherical co ordinate system, producing E radially in all the directions . Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 161

Potential due to Point charge
Assuming free space ,the field E due to a point charge Q at a point having radial distance r from origin is given by, Consider a unit charge which is placed at a point B which is at a radial distance of rB from the origin. The point A is at a radial distance of rA from the origin. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 162

Potential due to point charge
The differential length in spherical system is, Hence the potential difference VAB between points A and B is given by, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 163

Potential due to a point charge
Therefore we get, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 164

Absolute potential The absolute potentials are measured with respect to a specified reference positions. Such reference position is assumed to be at zero potential. SHIELDED CABLE Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 165

Absolute Potential Consider potential difference VAB due to movement of unit charge from B to A in a field of a point charge Q. Now let the charge is moved from infinity to the point A . i.e., rB = infinity. Hence, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 166

Absolute Potential Therefore, The potential of point A or absolute potential of point A is , The potential of point B or absolute potential of point B is, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 167

Absolute potential The potential difference can be expressed as the difference between the absolute potentials of two points. Hence the absolute potential at any point which is at a distance r from the origin of a spherical system is given by, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 168

Potential due to point charge not at origin
If the point charge is not located at the origin of a spherical system then obtain the position vector where Q is located. The absolute potential at a point A located at a distance r from the origin is given by, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 169

Potential due to a line charge
Line charge distribution Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 170

Potential due to a line charge

Potential due to surface charge
Surface charge distribution Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 172

Potential due to surface charge

Potential due to volume charge
Volume charge distribution Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 174

Potential due to volume charge

Potential Gradient The rate of change of potential with respect to the distance is called the potential gradient. Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 176

Gradient of a scalar In Cartesian coordinates, In Cylindrical coordinates, In Spherical coordinates, Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 177

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