Maxwell’s Equations – Differential form of Gauss’s Law For Electric Field(Part 2)


This is Part two of my eight part series on Maxwell Equations

In my previous article I discussed about Integral form of Gauss’s Law which is one of the Maxwell’s Equations. Now in this article I’ll discuss about Differential form of the Gauss’s Law.
The integral form of Gauss’s law for electric fields relates the electric flux over a surface to the charge enclosed by that surface. Like all of Maxwell’s Equations, Gauss’s law may also be written in differential form. So, Differential form of Gauss’s Law is
$\vec{\nabla }\cdot \vec{E}=\frac{\rho }{\epsilon _{0}}$
Now, differential form of Gauss’s law tells us how field behaves at a point which integral form of Gauss’s law can not tell , but both these laws describe same physical phenomenon.You can use any of the form depending on how useful that form is to the problem you are trying to solve.
The left hand side of the equation is the divergence of the electric field that is, the tendency of the field to ‘‘flow’’ away from a specified location , and the right side is the electric charge density divided by the permittivity of free space.

Main idea behind Gauss’s law in differential form is

The electric field produced by electric charge diverges from positive charge and converges upon negative charge.

Here we shall note that, the only places at which the divergence of the electric field is not zero are those locations at which charge is present. So, if positive charge is present then the divergence is positive which means that the electric field tends to ‘‘flow’’ away from that location. Now if negative charge is present then the divergence is negative and the field lines tend to ‘‘flow’’ toward that point.
Differential form of Gauss’s Law :- This form deals with the divergence of the electric field and the charge density at individual points in space.
Integral form of Gauss’s Law :- This form involves the integral of the normal component of the electric field over a surface.
If you know about spatial variation of the vector electric field is at a specified location then you can find the volume charge density at that location using this form. And if the volume charge density is known, then
divergence of the electric field may be determined.
So now you know about the essential difference between both the forms (note :- The differential form and the integral form are the same thing) I shall go ahead and describe each and every symbol used in mathematical form of this law.

  1. $\nabla $ the del operator :-  To know mathematics about this operator you can visit these two links (vector differentiation -1 and vector differentiation-2). Now I’ll try to explain what it is for in this law. Here presence of this operator tells you that you have to take the derivative of the quantity (here $\vec{E}$) on which it is acting. Here it is used with a dot ($\vec{\nabla }\cdot \vec{E}$) which indicates you have to take the divergence of the physical quantity.
  2. The divergence ($\vec{\nabla }\cdot$) :-  Now concept of divergence is very important when it comes to be about studying physics and engineering where you bother yourself with the behavior of vector fields.  Both flux and divergence deals with the flow of a vector field but there is an important difference that is  flux is defined over an area, while divergence applies to individual points.Now consider the case of flowing fluid here the divergence at any point is a measure of the tendency of the flow vectors to diverge from that point (that is, to carry more material away from it than is brought toward it). Thus points of positive divergence are sources (faucets in situations involving fluid flow, positive electric charge in electrostatics), while points of negative divergence are sinks (drains in fluid flow, negative charge in electrostatics).
  3. The divergence ($\vec{\nabla }\cdot \vec{E}$) of the electric field:- This expression represents the divergence of electric field in Gauss’s Law. In electrostatics as we already know all electric field lines begin on points of positive charge and terminate on points of negative charge, so it is obvious that this expression is proportional to the electric charge density at the location under consideration. So divergence at any point is simply not about spacing of field lines but it is about whether the the flux out of an infinitesimally small volume around the point is greater than, equal to, or less than the flux into that volume. So if flux in outward direction is greater than the inwards flux then the divergence is positive and if inward flux is greater than the outwards flux then the divergence is negative at that point. Also  if the outward and inward fluxes are equal the divergence is zero at that point.
  4. $\rho$ :-  It is the charge density in  coulombs per cubic meter

Now you know about the differential form of the Gauss’s Law and what makes it different from its integral form although conceptually both are same you can solve the problems related to this topic. You can even practice little mathematics where you can prove the equality of both the forms of Gauss’s Law.
There is still a lot of things remained that can be pointed out in this article but I intend to say it all when I prepare detailed noted on this topic.

Next article in this series would be about Integral form of Gauss’s Law for magnetic fields

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