coordinate systems & gauss s · vector field a vector field in the plane, can be visualized as...

COORDINATE SYSTEMS & GAUSS'S LAW

by

Dr. Sikder Sunbeam Islam

Dept. of EEE.

IIUC

Course Title: Engineering Electromagnetism

SYLLABUS (UP TO MID-TERM) : 30 MARKS

2

WHAT IS ELECTROMAGNETISM?

Electromagnetism is a branch of physics which deals

with electricity and magnetism and the interaction

between them.

Electromagnetism is basically the science of

electromagnetic fields. An electromagnetic field is the

field produced by objects that are charged electrically.

Radio waves, infrared waves, Ultraviolet waves, and

x-rays are all electromagnetic fields in a certain range

of frequency.

3

VECTOR BASICS

Vector: A quantity with both magnitude and

direction. Example: Force.

Scalar: A quantity that does not posses direction .

Example: Temperature.

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VECTOR BASICS :CONTINUES….

Vector Addition:

Vector Subtraction:

Vector Multiplication:

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Unit vectors: , and directed along x, y,

and z respectively with unity length and no

dimensions.

The vector r=A+B+C may be written in

terms of unit vectors as:

r =A+B+C = A +

Where:

A is the directed length or signed

magnitude of A.

Example: A unit vector in the direction of B is;


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Dot Product

Which results in a scalar value, and is the smaller

angle between A and B.


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Vector Basics :continues….

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Dot products of two vectors A and B is, 9


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Problem: a)Write the expression of the vector going from point

P1(1,3,2) to point P2(3,-2,4) in Cartesian coordinate. b) What is the

length of this line?


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VECTOR FIELD

A vector field in the plane, can be visualized as a

collection of arrows (flux lines) with a given magnitude

and direction, each attached to a point in the plane.

A vector filed strength is measured by the number of flux

lines passing through a unit surface normal to the

vector.

Flux flow of vector is like flow of water or fluid.

A enclosed surface of a volume will have

outward/inward flow of flux through this surface when

the volume contains source/sink.

The net outward flow per unit volume is therefore the

measure of strength of the enclosed source.

In the uniform field, there is an equal amount of inward

and outward flux going through any closed volume

containing no source or sink , results zero divergence.

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VECTOR FIELD

A kind of source called vortex source that causes

circulation of vector filed around it (source).

If A is a force acting on an object, its circulation

will be the work done by the force in moving the

object once around the contour.

Similarly, the phenomenon of water whirling down

a sink drain is an example of vortex sink.

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COORDINATE SYSTEM :

RECTANGULAR (CARTESIAN) COORDINATE

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COORDINATE SYSTEM :


A point is located by its and coordinates,

or as the intersection of three constant

surfaces (planes in this case) y , x, z 16

COORDINATE SYSTEM :


Increasing each coordinate

variable by a differential

amount dx, dy , and dz, one

obtains a parallelepiped. dx

dy dz

Differential volume: dv = dx dy dz

Differential Surfaces: Six planes with

dierential areas ds=dxdy; ds=dzdy; ds= dxdz

Differential length: from P to P’ ,

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Line Integral [Application of scalar(dot)product ] Suppose, we move along a path from P1 to P2 in a radial force field F. F acting in the r

direction. At any point P, the product of path length dL (incremental) and

component of F parallel to it is given by,

COORDINATE SYSTEM :


Component of dL in the r direction is, dr=cosϴ dL i.e.[cosϴ=dr/dL]

i.e.[cosϴ=FL/F]

Using vector (dot product),

If work dW done by force F moving an object a distance dr= cosϴ dL then,

dW=F.dL= Fcosϴ dL . Total work W,

[This formulation is called line integral.

[Work for path.]

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COORDINATE SYSTEM :


PROBLEM: Find the work required to move a 5KG mass from x=0,

y=0 to x=8, y=7 against a force . Ans=171J.

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Surface Integral:

COORDINATE SYSTEM :


We can express the flow of water, Ψ=BA cosθ=B.A (for Uniform)

Integrating the contribution of all

points across the surface of the loop,

obtaining the total flow of water ,

(for nonuniform)

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COORDINATE SYSTEM :


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Volume Integral: Example (Solve yourself)

COORDINATE SYSTEM :


Ans. 1440 kg

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COORDINATE SYSTEM :


The Divergence of A at a given point P is the outflow of flux

from a small closed surface per unit volume as the volume

shrinks to zero.

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COORDINATE SYSTEM :


The curl of A is an axial(rotational) vector whose magnitude is the

maximum circulation of A per unit area as the area tends to zero and

whose direction is the normal direction of the area.

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COORDINATE SYSTEM :


The divergence theorem states that the total outward flux of a vector

field A through the closed surface S is the same as the volume

integral of the divergence of A.

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The Stoke’s theorem proposing that the surface integral of the curl

of a vector field A over any surface bounded by a closed path is equal

to the line integral of a vector field A round that path.

COORDINATE SYSTEM :

CYLINDRICAL COORDINATE

[In some books is

expressed with r ]

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COORDINATE SYSTEM :

CYLINDRICAL COORDINATE

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COORDINATE SYSTEM :

SPHERICAL COORDINATE

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COORDINATE SYSTEM :

SPHERICAL COORDINATE

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COORDINATE SYSTEMS

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COORDINATE SYSTEM : TRANSFORMATIONS BETWEEN

CYLINDRICAL AND CARTESIAN COORDINATES

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OR,

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Example-2: Transform the vector

in Cartesian Coordinates.

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SPHERICAL AND CARTESIAN COORDINATES

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36



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Electrostatic Field

Topics to be covered

(Ref. Chapter-4)

• Article 4.1-4.4 (Basic concept of Electric field)

• Article 4.5-Gauss’s law specifically Maxwell’s equation

• Article 4.6-Application of Gauss’s Law

• Article 4.7-Electric Potential

• Article 4.8- Relationship between E & V

• Article 4.9-An Electric Dipole & Flux Lines

• Article 4.10-Energy Density in Electrostatic Fields

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Basic concept of Electric field

Electric charge (Q) is the physical properties of matter

that cause it to experience a force when placed in an

electromagnetic field.

Point Charge means a charge that is located on a body

whose dimensions are much smaller than the relevant

dimension.

Coulomb’s Law: This law states that, two point charge

Q1 and Q2 separated by a distance R experience a force,

So,

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Basic concept of Electric field: cont….

Electric Field Intensity (E) is the force per

unit charge when placed in the electric field.

So,

Electric Fields due to continuous charge distribution

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Electric Flux Density (D)

Basic concept of Electric field: cont….

So,

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GAUSS’S LAW-MAXWELL’S EQUATION

Gauss’s law states that, the total electric flux through

any closed surface is equal to the total charge enclosed by the

surface.

-------------(1)

Charge density

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Now applying Divergence theorem ( ) to equ.(1),

GAUSS’S LAW-MAXWELL’S EQUATION

-------------(2)

Now comparing eqt.(1) and (2) we find,

-------------(3)

Which is the first of four Maxwell’s Equations to be derived. It states

that a volume charge density is the same as Divergence of Electric

flux density.. Is the charge per unit volume.

1. Equation 1 & 3 states the same Gauss’s law in different way (equ.1 is

integral form whereas equ.3 is differential form)

2. Gauss's law is an alternative statement of Coulomb's law; proper application

of the divergence theorem to Coulomb's law results in Gauss's law.

3. Gauss's law provide* an easy means of finding E or D for symmetrical

charge distributions such as a point charge, an infinite line charge, an infinite

cylindrical surface charge, and a spherical distribution of charge.

Points to be noted

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Gauss’s law-Maxwell’s equation

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APPLICATIONS OF GAUSS'S LAW

Gauss's law to calculate the electric field

involves first knowing whether symmetry

exists.

If field is symmetric, we construct a

mathematical closed surface (known as a

Gaussian surface).

The surface is chosen such that D is normal or

tangential to the Gaussian surface.

if D is tangential, then D. dS=D.ds, as D is

constant on the surface.

if D is normal, then D. dS=0.

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APPLICATIONS OF GAUSS'S LAW: POINT CHARGE

Suppose, a point charge Q located at the origin of spherical

coordinate system. To determine D at a point P it is easy to assume a

closed surface of radius r=a containing P.

-------------(1) At, r=a

P

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APPLICATIONS OF GAUSS'S LAW: UNIFORMLY

CHARGED SPHERE

-------------(1)

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CHARGED SPHERE-CONT….

or,

while,

-------------(2)

-------------(3)

-------------(4)

Now from equ.4 and 5,

-------------(5) 48



-------------(6)

Now from equ.3 and 6,

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50


Example:

or,

Apply Gauss Law,

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Example:

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ELECTRIC POTENTIAL

-------------(1)

-------------(2)

Dividing W by Q gives potential energy per unit charge

Denoted by as potential difference between A and B points.

-------------(3)

If E field in Fig. is due to point charge Q located at origin, then -------(4)

Now from, equ.3

Then, Potential,

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ELECTRIC POTENTIAL: EXAMPLE

Hence,

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RELATIONSHIP BETWEEN E AND V-MAXWELL’S

EQUATIONS

Now we know, the potential difference between points A and B is independent

of the path taken. Hence,

-------------(1)

This shows that, the line integral of E along a closed path must be zero. Physically it

reveals that no net work is done in moving a charge along a closed path in electrostatic

field. Applying Stokes’s Theorem in equ.(1) ,

-------------(2)

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Vectors whose line integral does not depends on the path of integral are called

conservative vectors. Thus Electrostatic field is a conservative field. Equ.

(1) or (2) referred to as Maxwell’s Equation. Equ(1) is integral form and

equ(2) id differential form.

RELATIONSHIP BETWEEN E AND V-MAXWELL’S

EQUATIONS

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Example:

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AN ELECTRIC DIPOLE & FLUX LINES

When two point charges of equal magnitude but opposite sign are

separated by a small distance then an Electric Dipole is formed.

When r1 and r2 is the distance

between P and +Q and –Q

respectively. If dipole moment

is p then, the potential at

origin is,

An electric flux line is an imaginary path or line drawn in

such a way that its direction at any point is the direction of

the electric field at the point.

Note that the dipole moment p is directed from -Q

to +Q. If the dipole center is not at the origin but at

r', then equtn. Becomes,

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EXAMPLE

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ENERGY DENSITY IN ELECTROSTATIC FIELDS

Suppose we wish to position three point charges Q1, Q2, and Q3

in an initially empty space shown shaded in Figure.

Hence the total work done in positioning the three charges is,

where V1, V2, and V3 are total potentials at P1, P2, and P3, respectively.

In general, if there are n point charges, eq. (1) becomes

Fig. Assembling of

charges

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ENERGY DENSITY IN ELECTROSTATIC FIELDS

According to Eqn.(2),

So, from Eqn.(4),

By applying divergence theorem to the first term at right side

while dS varies as . Consequently, the first integral in eq. (6) must tend to zero as

the surface S becomes large. Hence, eq. (6) reduces to

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EXAMPLE

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Alternatively

REFERENCE

Engineering Electromagnetics; William Hayt &

John Buck, 7th & 8th editions; 2012

Electromagnetics with Applications, Kraus and

Fleisch, 5th edition, 2010

Elements of Electromagnetics ; Matthew N.O.

Sadiku

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