Want to know about divergence 
*

*Can anyone please explain how to know whether a vector field has divergence or not by seeing its diagram? 

*I have read that a vector field must change for having divergence but why is divergence zero in middle of electric field between dipole?
 A: *

*Divergence can be thought of as the flux of a vector field per unit volume. It is positive if there is a net flux out of a small volume and negative if there is a net flux inwards.


When you say "its diagram" - of course there are different ways of plotting vector fields.
Perhaps the most common way is using field lines. In which case it can be straightforward to recognise places where there is non-zero divergence because field lines would either begin or end.
For example consider a point positive charge with a set of radial field lines going away from the charge (picture from http://dc458.4shared.com/doc/M0envQzJ/preview.html). The electric field lines originate on the positive charge. Therefore the divergence at the charge must be non-zero and positive.
At any point in space away from the charge, field lines neither begin nor end and just stretch away to infinity. This is telling you that the divergence at all other points in space is zero. Doing the Maths will confirm this is true.
Vice-versa, field lines end on a negative charge and so the divergence is negative.

I think it is much more difficult to judge whether there is divergence when fields are represented by arrows of length proportional to magnitude. However, the same principle applies; you are looking for whether the net flux of the vector field from a volume is non-zero.


*Let's use the electric fields of a dipole as an example. Here is the plot using a field line representation.



I hope it's clear that there are only two places in the diagram where field lines are either beginning or ending and that is on the charges. Field lines are "created" on the positive (red) charge and "destroyed" on the negative (blue) charge. The divergence at these points is positive or negative respectively.
Elsewhere in space the field lines are continuous and the divergence is zero (which is of course in accordance with Gauss's law that $\nabla \cdot {\bf E} = \rho/\epsilon_0$). In particular, there is no divergence at the point midway between the two charges because you can visually confirm that no field lines begin or end there and mathematically because $\rho =0$ there.
Now consider a different representation of these fields - with small arrows.

It is fairly obvious (to me) that the divergence is non-zero where the charges are, but I think it is much harder to tell elsewhere in the plot that the net flux into or out of a very small volume is zero.
