Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

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Electrical Engineering (EE) : Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

The document Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Electromagnetic Theory.
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Till now we have talked about operators such as gradient, divergence and curl which act on scalar or vector fields. These operators are all first order differential operators. Gradient operator, acting on a scalar field, gives a vector field. Divergence, on the other hand, acts on a vector field giving a scalar field. The curl operator, acting on a vector field, gives another vector field.  The Cartesian expressions for these operators are as follows : The operator  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

where we have denoted a scalar field by f and a vector field by Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
We will define another useful operator, known as the Laplacian operator, which is a second order differential operator acting on a scalar field (and with some conventional usage on a vector field). This operator is denoted by  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev i.e. it is a divergence of a gradient operator. Since the gradient operates on a scalar field giving rise to a vector, the divergence operator can act on this finally resulting on a scalar field. Thus

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

The operator  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev denoted by Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev as the Laplacian.

A class of functions, known as “Harmonic Functions” satisfy what is known as the Laplacian equation,

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

In electromagnetic theory, in particular, one often finds Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev operator acting on a vector field. As has been explained above, a Laplacian can only act on a scalar field. However, often we have equations where the Laplacian operator acts on components of a vector field, which are of course scalars. Thus Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev is used as a short hand notation, which actually means Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev where Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev are the unit vectors along three orthogonal directions in the chosen coordinate system and Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev are the components of the vector field  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev  directions. Thus, in Cartesian coordinates, we haveElements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

In electrodynamics, several operator identities using the operator  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev frequently used. Here is a list of them. They are not proved here but you are strongly advised to prove some of them.

1.  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev This is obvious because Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev represents a conservative field, whose curl is zero.

2.  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev We have seen that Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev represents a solenoidal field, which is divergenceless. We will see later that the magnetic field B is an example of a solenoidal field.

3.  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev This operator identity is very similar to the vector triple product. (for ordinary vectors, we have  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

4.  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev where both f and are scalar fields.

5.  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

6.  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Green’s Identities :

We will now derive two important identities which go by the name Green’s identities.

Let the vector field Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRevbe single valued and continuously differentiable inside a volume V bounded by a surface S. By divergence theorem, we have

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

If we choose Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev are two scalar fields, then we get, using the relation (5) above,

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

This is known as Green’s first identity. By interchanging Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Equation (3) is Green’s second identity and is also known as the Green’s Theorem.

Uniqueness Theorem :

An important result for the vector fields is that in a region of volume V defined by a closed surface S, a vector field is uniquely specified by

1. its divergence
2. its curl and
3. its normal component at all points on the surface S.

This can be shown by use of the Green’s theorem stated above. Consider two vector fields Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev which are specified inside such a volume and let us assume that they have identical divergences, curl as also identical values at all points on the defining surface. The uniqueness theorem implies that the two vector fields are identical.

To see this let us define a third vector Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev By the properties that we have assumed for  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRevElements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev that curl of  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev is a conservative field and hence can be expressed as a gradient of some potential. Let Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev The equality of the divergences of Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev implies that the divergence of  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev which, in turn, implies that Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev the scalar potential  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev the Laplace’s equation at every point inside the volume V.

The third property, viz., the identity of the normal components of Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev at every point on S implies that the normal component of Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev Let us use Green’s first identity , given by eqn. (1) above taking Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

the last relation follows because  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev everywhere on the surface. Thus we have,

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Since  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev everywhere on the surface, it gives  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev Since an integral of a function which is positive everywhere in the volume of integration cannot be zero unless the integrand identically vanishes, we get Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev proving the theorem.

Dirac Delta Function

Before we conclude our discussion of the mathematical preliminaries, we would introduce you to a very unusual type of function which is known as Dirac’s δ function. In a strict sense, it is not a function and mathematicians would like to call it as “generalized function” or a “distribution”. The defining properties of a delta function are as follows :

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

In the last relation, it is to be noted that the limits of integration includes the point where the argument of the delta function vanishes. Note that the function is not defined at the point x=0. It easily follows that if we take a test function f(x) , we have

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

We can easily see why it is not a function in a strict sense. The function is zero at every point other than at one point where it is not defined. We know that a Riemann integral is defined in terms of the area enclosed by the function with the x axis between the limits. If we have a function which is zero everywhere excepting at a point, no matter what be the value of the function at such a point, the width of a point being zero, the area enclosed is zero. As a matter of fact, a standard theorem in Riemann integration states that if a function is zero everywhere excepting at a discrete set of points, the Riemann integral of such a function is zero.

How do we then understand such a function? The best way to look at a delta function is as a limit of a sequence of functions. We give a few such examples.

1. Consider a sequence of functions defined as follows :

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

For a fixed n, it represents a rectangle of height n, spread from Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev The figure below sketches a few such rectangles.

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

As n becomes very large, the width of the rectangle decreases but height increases in such a proportion that the area remains fixed at the value 1. As n → ∞ the width becomes zero but the area is still finite. This is the picture of the delta function that we talked about.

2. As second example, consider a sequence of Gaussian functions given by

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

These functions are defined such that their integral is normalized to 1,

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

for any value of σ. For instance, if we take σ to be a sequence of decreasing fractions 1,1/2,

…,1/100, .. the functions represented by the sequence are Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRevwhich are fast decreasing functions and the limit of these when σ approaches zero is zero provided x ≠ 0. Note, however, the value of the function at x=0 increases with decreasing σ though the integral remains fixed at its value of 1.

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

3. A third sequence that we may look into is a sequence of “sinc” functions which are commonly met with in the theory of diffraction  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev Note that as ∈  becomes smaller and smaller, the width of the central pattern decreases and the function gets peaked about the origin. Further, it can be shown that  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev Thus the function Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev is a representation of the delta function in the limit of Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

This gives us a very well known integral representation of the delta function

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
One can prove this as follows:

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

where we have set Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev to arrive at the last result. It may be noted that the integrals above are convergent in their usual sense because sin (nx) does not have a well defined value for Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

4. An yet another representation is as a sequence of Lorentzian functions  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev It can be seen that Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev Some properties of δ- functions which we state without proof are as follows:

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

We end this lecture by summarizing properties of cylindrical and spherical coordinate systems that we have been using (and will be using) in these lectures.

Cylindrical Coordinate system

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

The unit vectors of the cylindrical coordinates are shown above. It may be noted that the z axis is identical to that of the Cartesian coordinates. The variables ρ and θ are similar to the two dimensional polar coordinates with their relationship with the Cartesian coordinates being given by  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev An arbitrary line element in this system is  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Another important quantity is the volume element. This is obtained by multiplying a surface elent in the ρ−θ plane with dz. And is given by Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Spherical Coordinate system

For systems showing spherical symmetry, it is often convenient to use a spherical polar coordinates. The variables  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev and the associated unit vectors are shown in the figure.
Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRevElements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

To define the coordinate system, choose the origin to be at the centre of a sphere with the Cartesian axis defined, as shown. An arbitrary point P is on the surface of a sphere of radius r so that OP=r, the radially outward direction OP is taken to be the direction of the vector  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev two coordinates are fixed as follows. The angle which OP makes with the z axis is the polar angle q and the direction perpendicular to OP along the direction of increasing θ is the direction of the unit vector Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev varies from 0 to π. The azimuthal angle φ is fixed as follows. We drop a perpendicular PP’ from P onto the xy plane. The foot of the perpendicular P’ lies in the xy plane. If we join OP’, the angle that OP’ makes with the x-axis is the angle φ Note that as the point P moves on the surface of the sphere keeping the angle θ fixed, i.e. as it describes a cone, the angle φ increases from a value 0 to 2π. The relationship between the spherical polar and the Cartesian coordinates is as follows : 

The volume element in the spherical polar is visualized as follows :

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

The expression for the divergence and the Laplacian in the spherical coordinates are given by

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Delta function in three dimensions :

In three dimensions, the delta function is a straightforward generalization of that in one dimension 

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

With the property

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Where the region of integration includes the position Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Example : In electrostatics we often use the delta function. In dealing with a point charge located at the origin, one comes up with a situation where one requires the Laplacian of the Coulomb potential, i.e. of 1/r. In the following we obtain Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Since the function does not have angular dependence we get, by direct differentiation, for r ≠ 0,

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

This is obviously not valid for r=0. To find out what happens at r=0, consider, integrating Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev over a sphere of radius R about the origin. R can have any arbitrary value. We have using, the divergence theorem,
Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

where the surface integral is over the surface of the sphere. Since, r=R on the surface of the sphere (and is non-zero),Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev on the surface. We then have

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

This implies that Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Tutorial Problems :

1. Calculate the Laplacian of Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
2. Obtain an expression for the Laplacian operator in the cylindrical coordinates.
3. Obtain the Laplacian of  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
4. Show that  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
5. Find the Laplacian of the vector field  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Solutions to Tutorial Problems :

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
2. In cylindrical coordinates, the z-axis is the same as in Cartesian. Thus we need to only express Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev in polar coordinates and add Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev to the result. We have Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev We can thus write,

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

One can similarly calculate  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev and show that it is given by
Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Adding these terms and further adding the term Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

3. In Cartesian coordinates Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev where we have used Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

The second derivative with respect to  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev adding the second differentiation with respect to y and z (which can be written by symmetry), we get

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Where we have used Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

4.  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Adding three components result follows.

5. The Laplacian of a vector field is

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

Self Assessment Quiz

1. Evaluate Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
2. Find the Laplacian of  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
3. Determine Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
4. Evaluate Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

 

Answer to Self Assessment Quiz

Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRevElements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRevElements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev we get the result to be Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
2. Answer :  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
3. 3. Only the delta function at x=0 contributes because the argument of the first delta function  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev is not in the limits of integration. The result is Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev
4. In this case Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev is inside the limit of integration. Thus the result is  Elements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRevElements of Vector Calculus: Laplacian Electrical Engineering (EE) Notes | EduRev

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