lecture1.1

Electromagnectic Field Theory

PH-207

B.Tech 4

Semester

NIT Jalandhar

https://sites.google.com/site/karvindk2013

/

Contents of Course:

Chapter I: Electrostatic Fields

Chapter II:

Magnetic Fields

Chapter III: Maxwell Equations

Chapter IV:

Electromagnetic Waves

Chapter V: Poynting Vector and Flow of Power

Chapter VI:

Guide Waves and Waveguides

Books Recommended:

1. Field and Wave Electromagnetics by David K. Cheng

2.

Engineering Electromagnetics

by W H Hayt & J A Buck

3. Elements of Electromagnetics

by Matthew N.O Sadiku

4. Electromagnetic Waves and Radiating System by Jordan

and Balmain

Electromagnectic Field Theory

PH-207

Chapter I

Electrostatic Fields

Orthogonal Co-ordinate system: In three dimensional space a point can be located as the intersection of three surfaces. Suppose these

three surfaces are described by u₁ = constant, u₂ = constant,

u₃ = constant . All u need not be length. When three surfaces are

perpendicular to each other, we call it orthogonal co-ordinate systems.

Examples: Cartesian co-ordinate system, Cylindrical co-ordinate system and spherical co-ordinate system.

Let are unit vector along three orthogonal directions. These unit vectors are also called base vectors.

As the vectors are perpendicular to each other so

A vector A can be written in component form as,

Magnitude of above vector is

Differential length element is written as

Total differential change in arbitrary direction is

Above equation can further be written as

Differential area whose direction is normal to unit vector a_u1 is written as

Differential area whose direction is normal to unit vector a_u2 is written as

Gradient of scalar field: The gradient of a scalar field is a vector which represent the magnitude and direction of the maximum

space rate of increase of scalar.

---(1)

In general orthogonal curvilinear

co-ordinate system, we can write the gradient of scalar as,

---(2)

In Cartesian co-ordinate system, (u1, u2, u3) (x,y,z) and h₁ = h₂ = h₃ = 1.

---(3)

In above equation V is the scalar field and ‘ ’ is del operator

---(4)

Note from equation (1) the gradient of scalar field gives us vector quantity. Note that a_x, a_y and a_z are the unit vectors in Cartesian Co-ordinate system along x, y and z-directions respectively.

Divergence of vector field: The divergence of the vector field say A about a point is defined as the amount of outward flux per unit

volume as the volume about that point tends to zero

---(1)

We can write the divergence of vector field A in Cartesian co-ordinate system can be written as

= ---(2)

In general curvilinear co-ordinate system, we write the divergence as

Curl of vector field: The curl of a vector field is a vector whose magnitude is the maximum net circulation of A per unit area as the area tend to zero and whose direction is the maximum direction of area when the area is oriented to make the net circulation maximum.

---(1)

In Cartesian co-ordinate system the curl of vector field A can be written As

---(2) Above Equation can be expressed in determinant form

In general orthogonal curvilinear co-ordinate system, we write

---(4)

Using the appropriate values of u₁, u₂ and u₃ and metric coefficients h₁, h₂, h₃ we can write the curl in Cartesian cylindrical or spherical co-ordinate system.

Whirlpool in pond, an example of

Fundamental Theorem of calculus:

Consider a function f(x) of one variable x. The fundamental theorem of calculus states that

---(1)

We write the above Eq. as follows:

Above Eq. says that if we want to find the integral of some function F(x) then we just need to find a function f(x) such that = F(x) and then the difference of the values of function f(x) at end points ( f(b)-f(a)) will give us the integral of the function F(x).

Fundamental Theorem of gradients:

We consider a scalar function T(x,y,z) which depends upon three variables. We start from some point say a and moves through displacement vector dl₁. The change in the scalar function T during this

displacement is given by

Suppose we further moves through dl₂ and then dl₃ and So on.

The total change in T as we move from point a to b is given by the integral

Above Eq. gives us the fundamental theorem of gradients. It states that the

Note the following points:



Fundamental theorem of divergence:

Divergence theorem: The volume integral of the divergence of the vector field A equals to the total outward flux of the vector field through the surface which bound the volume.

---(1)

Proof: To prove above theorem we consider the small volume element bounded by the surface s_j. Using the definition of the divergence of the vector field, we can write

Consider an arbitrary volume element V divided into N small differential volume element .

Combining the contribution of all small volume elements and using eq (2), we write,

---(4)

Left side of above eq. Can be written is actually the volume integral fo the divergence of the vector field,

The surface integral in equation (3) involve the summation over all the faces which bound the all small volume elements. The contributions of internal faces adjacent to each other will cancel out. The reason is that at a common internal interface the outward normal of adjacent

element points in opposite directions. Hence the net contribution of right side of eq (3) comes only from the external surface S bounding the whole volume V

---(5)

Using (4) and (5) in Eq (3), we get

---(6)

Fundamental theorem of curl:

Strokes' Theorem: The surface integral of the curl of the vector field over the open surface is equal to the closed line integral of the

vector field along the contour bounding the surface.

---(1)