What is the Divergence of a vector field?
Technically the Divergence at the given point is defined as the net outward flux per unit volume as the volume shrinks (tends to) zero at that point. For the vector field
In simple words, it represents whether the field lines are spreading or compressing at the given point present in the field. Or again simplified, it is the term to represent the “Outgoingness” of the field around that point. The following article explains the definition of the Divergence of a vector field in detail.
The Divergence Symbol
The divergence operation is applied to the vector function/field. Let us say the vector field is denoted as A then the divergence symbol or the representation is as follows-
The Divergence Formulas in different coordinate systems
Divergence of a vector field in Cartesian Coordinates
Divergence in Cylindrical Coordinates
Divergence in Spherical Coordinates
Intuitive derivation of the Divergence formula
The divergence indicates the outgoingness of the field at the point of interest. Let vector field A is present and within this field say point P is present. The vector field means I want to say the given vector function of x, y and z. I am assuming the Cartesian Coordinates for simplicity. So the field is A(x,y,z). Let us say we want to analyze the divergence of the field around this point P.
Now as explained in the last article, the divergence can be thought as the net number of lines or net flux of the field coming out of the infinitesimal volume supposed at the given point. Isn’t it? So let us assume the very small or infinitesimal volume in the form of the
As we know from the knowledge of the Cartesian Coordinate System that volume can be formed by varying all three lengths i.e. along X, Y and Z axes. So let us say our infinitesimal volume have the edges dx, dy and dz. In other words, it would have the volume dv = dx dy dz. Point P is in the heart of the cube.
The vector field A is present everywhere and hence at P also. Let the coordinates of point P are (a,b,c) and value of the function/field at this point P (a,b,c) is A(a,b,c).
Now, you might be knowing about the Taylor Series. It is a mathematical expansion series of the function around the specific point. In other words, by using Taylor Series expansion, we actually look at the function from the perspective of the given point. We express the given function using the value of the function at that specific point and variation of the function (of course using derivatives) around that point. Actually, the reason behind this discussion is we are going to express our vector field/function using Taylor Series at point P.
For simplicity consider only x component of the field. Let us leave the y and z components, which can be easily be written by analogy. So with the use of Taylor Series expansion, the x component of the given function A can be expressed as below. Note that we are just representing the given function with another way. This wouldn’t change anything of the original function. We are doing so for our ease of calculations.
On the similar lines the Ay and Az components can be expressed for the given function A. We can also neglect the higher order derivative term as we are considering a very small volume. So we have just represented our given vector field by different presentation i.e. by using the series. With this representation, you can see that the function is reframed as its value at point P(a,b,c) and change of the function along X, Y and Z-axis from that point.
Now let us recall that the divergence is nothing but the net flux of the vector field that is coming out from the small volume (close surface) formed around P. Now how do we calculate the flux of the field? Recall the surface integration. We find the flux using the following formula. Here A is the vector field and ds is the infinitesimal surface of the given surface from which the flux is to be calculated.
Now for our case, we can neglect the integration as we are already assuming the infinitesimal case. So in simple words for our case flux would be the multiplication of the vector field and infinitesimal area.
Now as seen from the figure above, if I want to calculate the total flux that is coming out of this cube then I have to apply the flux formula for six times. Each time for a single surface, isn’t it? But for simplicity, we have considered only the x-component of the given field. So as you can see from the figure, we will calculate the flux only for the left and right surfaces. The x-components of the given field would go tangential to top, bottom and side surfaces. So the flux contribution for these surface would be zero for the Ax.
So let us take the right side surface. The area of this surface is
From the above expression, at very first, we can omit the higher derivative terms as we are dealing with the infinitesimal case. Secondly, the derivative terms with respect to y and z can also be neglected, as for the simplicity, we are considering only the x components of the given field i.e. Ax. And the y and z components of the field are not going to contribute to the flux of this surface as they will flow tangentially to it. So the above formula can be narrowed down to this-
Note here the value of the x at the position of the surface is a+(dx/2) as seen from the figure. So,
So the flux through this right surface is given as follows by multiplying the Ax with the area of the surface.
Now for the left side surface, a value of the x is a–(dx/2). Also, the ds value would be (-dydz) for taking the account of the direction of the area vector always out of the volume and hence pointing to the negative X-direction. So putting on the same expression above, we will get the flux for this left side surface as shown below.
So the net flux through these left and right surfaces is obtained by adding both of the above expressions which would lead to as shown below.
The y component of the field i.e. Ay comprises the flux linked to the top and bottom surfaces. While the z component Az make up the flux through the front and back ones. So working on the similar lines flux through these surfaces can be written as follows:
So considering all the surfaces i.e. the net flux coming out of this infinitesimal volume is given as:-
Now again recall the definition of the divergence. Let me state it once again in simple words, it is the total flux per unit volume. Right? Check the above expression, the term
The same expression can be modified for the cylindrical and spherical coordinates by converting the variables from cartesian to respective coordinates and obtained as stated above. Can you comment on how do we get such conversions?
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