# What is the Divergence of a vector field?

The Divergence of a vector field is a measure of the net flow of the flux around a given point. It is a basic term and used in many terminologies of Electromagnetics. This article defines the divergence

## A formal definition of Divergence

The divergence of vector field at a given point is the net outward flux per unit volume as the volume shrinks (tends to) zero at that point.

### Its meaning in simple words

Consider any vector field and any point inside it. Let us assume an infinitesimally small hypothetical volume around the considered point.

If the lines of the field that are going outside the hypothetical volume are more than that of coming inside it then we can say that the

The simplest synonym for the divergence is “outgoingness”. More is the divergence more are the field lines coming outside around that point and it should be considered as positive divergence. In contrast to that, if field lines tend to come inside then we can say the field is accumulating at that point which is opposite to “outgoingness” and hence considered as a negative divergence.

## The divergence explained in detail

In simple words, it would tell us about the nature of the

Consider again the vector field and let us say we are interested in finding the divergence at point P as shown in the figure above. Consider the infinitesimally small close surface preferably a cube around the point P having volume ∆v.

*Click here to read about the close surface.*

We are intentionally assuming the closed surface around the point so that we could find the flux of the field that is coming out of this

The reason is quite simple! We can infer about the outgoingness of the field by counting the total number of field lines that are coming outside this close surface versus the total number of lines that are coming inside of it. And the best way to predict the lines of the field is the flux.

Now how to find the net flux coming out of this cube? Simple, using surface integration. We have to apply the surface integration for each surface (considering the direction of unit vector outward) of the cube and every time the answer will be the total flux coming out of that respective surface. Combining all, the total close surface integration will give us the net flux that is coming out of the cube/close surface.

Now, if the answer to this net flux is positive then we can say a few lines of the field are coming outside more than that are coming inside. Hence the field is diverging.

If the net flux is negative then we can say that there are more lines coming into the cube than leaving hence converging.

Finally, if the

So all the necessary properties that define the divergence can be extracted from this net flux coming out of the infinitesimal (very small) close surface around the given point. Precisely we should keep this close surface as small as possible. In other words, its volume should be approaching zero. Hence finally the divergence can be defined and represented as below.

*Suggested Read:**Surface Integration**Suggested Community: Electromagnetics for GATE & ESE*

Tag:Electromagnetism