Lceture Set 4-Dell Operator

ET322 Dell operator 2014/kyu

DEL OPERATOR

The Del operator, written V, is the vector differential operator. In Cartesian coordinates,

This vector differential operator, otherwise known as the gradient operator, is not a vector in

itself, but when it operates on a scalar function, for example, a vector ensues. The operator is

useful in defining,

1. The gradient of a scalar V, written, as V

2. The divergence of a vector A, written as • A

3. The curl of a vector A, written as X A

4. The Laplacian of a scalar V, written as 2V

Likewise, the dell operator can be defined in other coordinate systems as

In cylindrical system and it spherical system, the operator is defined as,

Gradient of a scalar

The gradient of a scalar field V is a vector that represents both the magnitude and the direction of

the maximum space rate of increase of V.

The gradient of a scalr field V can be expressed in Cartesian coordinates as,

In cylindrical coordinates, the Gradient can be expressed as,


In spherical coordinates, the gradient is expressed as

Take note of the following properties of the scalar field V.

1. The Magnitude of V equals the maximum rate of change in V per unit distance.

2. V points in the direction of the maximum rate of change in V.

Example

Divergence of a vector and divergence theorem

We now define the divergence of a vector field A as the net outward flow of flux per unit volume

over a closed incremental surface.

The divergence of A at a given point P is the outward flux per unit volume as the volume

shrinks about P.

Hence,


where V is the volume enclosed by the closed surface S in which P is located. Physically, we

may regard the divergence of the vector field A at a given point as a measure of how much the

field diverges or emanates from that point. Figure (a) shows that the divergence of a vector field

at point P is positive because the vector diverges (or spreads out) at P. In Figure (b) a vector field

has negative divergence (or convergence) at P, and in Figure (c) a vector field has zero

divergence at P.

Divergence expressions.

Just like the Gradient, the divergence of a vector field A can be expressed in the three coordinate

systems.

In Cartesian coordinate systems,Div A is given as,

In cylindrical system, Div A is given as,

In spherical coordinates,


Divergence theorem,

From the definition of the divergence of A , it is not difficult to expect that

This is the divergence theorem

Statement: 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.

Example


Exercise


CURL OF A VECTOR AND STOKES THEOREM

The curl of A is an axial (or 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

when the area is oriented so as to make the circulation maximum.''

That’s

where the area S is bounded by the curve L and an is the unit vector normal to the surface

S and is determined using the right-hand rule.

In Cartesian coordinate,

Which simplifies to

By transforming the above equation using point and vector transformation techniques seen

earlier, we obtain the curl of A in cylindrical coordinates as


Which simplifies to

In spherical coordinates

Which simplifies to

Note the following properties of the Curl

The physical significance of the curl of a vector field is evident in its defining equation; the curl

provides the maximum value of the circulation of the field per unit area (or circulation density)

and indicates the direction along which this maximum value occurs. The curl of a vector field A


at a point P may be regarded as a measure of the circulation or how much the field curls around

P. For example, Figure (a) shows that the curl of a vector field around P is directed out of the

page. Figure (b) shows a vector field with zero curl.

Also, from the definition of the curl of A ,we may expect that

This is stoke’s theorem

Stokes's theorem states that the circulation of a vector field A around a (closed) path l is equal

to the surface integral of the curl of A over the open surface S bounded by l provided that A and

X A are continuous on S

Example: Determine the curl of the vector fields in the previous example.


LAPLACIAN OF A SCALAR

The Laplacian of a scalar field V, written as 2V is the divergence of the gradient of V.

In Cartesian coordinates

Notice that the Laplacian of a scalar field is another scalar field.

In cylindrical coordinates


In spherical coordinates

A scalar field V is said to be harmonic in a given region if its Laplacian vanishes in that region.

In other words, if 2 0V

Since the Laplacian operator 2 is a scalar operator, it is also possible to define the Laplacian of

a vector A. In this context, 2 A should not be viewed as the divergence of the gradient of A,

which makes no sense. Rather, 2 A is defined as the gradient of the divergence of A minus the

curl of the curl of A

That is,

Example


Exercise :

Classification of vector fields

A vector field is said to be;

1. Solenoidal (Divergence less) if . 0A


2. Irrotational (potential/Conservative ) if 0XA

MAXWELL’S EQUATIONS AS GENERALISATIONS OF CIRCUIT EQUATIONS

Maxwell’s equations can be obtained as generalizations of Ampere’s, Faraday’s, and Gauss’s

laws, which are circuit equations.

(i) Ampere’s law: dl I 1.8

Note: A Capacitor stores energy predominantly in the electric field while an Inductor stores

energy predominantly in the magnetic field.

Stokes theorem coverts the line integral in equation 1.8 around a closed path to an integral over

the surface enclosed by the path. Consequently, a more general relation is obtained by

substituting for I using the conduction current density, J. An even more general expression is

obtained by including the displacement current density, D t to give:

s s s

D Ddl J ds ds J ds

t dt

1.9

This is the loop or mesh form of one of Maxwell’s equations derived from Ampere’s law. Using

Stoke’s theorem, LHS of the integral in equation 1.9 can be converted to an open surface

integral. We thus get the point form of the equation:

DJ

t

1.10

(ii) Faraday’s Law (for constant flux): d

dt

1.11

Where; V is the induced emf in a circuit and is the total magnetic flux linking the circuit.

Since voltage is the integral around the circuit of ,dl and is the integral of ds over the

surface enclosed by the circuit, the more general form of equation 1.11 is:

s

dl dst

1.12

The surface may be changing so the time derivative should be inside the integral sign. This is

another one of Maxwell’s equations. The point relation is obtained by applying Stokes theorem

to get:


t

1.13

(iii) Gauss’s law (electric field) D ds Q 1.14

Generally, total charge is the integral, over the volume of interest, of the charge density, p.

Equation 1.14 becomes:

D ds dv 1.15

The relation is obtained by applying the divergence theorem (which converts an integral over a

closed surface to a volume integral within the volume enclosed) to the LHS of equation 1.15 to

give:

D 1.16

(iv) Gauss’ law (magnetic field) 0ds 1.17

The magnetic field does not have source points. Thus, there is no such things as a magnetic

charge, implying that magnetic charge = 0 as in equation 1.7. Applying the divergence theorem

gives

0 1.18

To summarize these results:

0

J D

D

0

dl D J ds

dl ds

D ds dv

ds

I

II

III

IV

The above field equations have been obtained as generalizations of circuit equations. These four

equations contain the continuity equation,

J or J ds dv 1.19

Power and poyntig vector

As mentioned before, energy can be transported from one point (where a transmitter is located) to another point

(with a receiver) by means of EM waves. The rate of such energy transportation can be obtained from Maxwell's

equations

Lceture Set 4-Dell Operator

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