Poisson's Equation And Its Meaning

Whenever, it comes to some basic and important equations in Electrostatics, Poisson's Equation always makes it to the the first one to be named. It is one of the key equations in Electrostatics and thus, is the subject of our second article in the Month of Equations Series.

Meaning of equation :

Poisson's equation basically states that:

The potential distribution in a region depends upon the region's local charge distribution.

Poisson's Equation
Here, V presents the Electrical Potential, Rho represents the charge density/local charge distribution & Epsilon represents the permittivity.

This equation involves the use of Laplacian operator in it. So, first of all, we need to know what actually the Laplace operator signifies.

Laplacian operator

Named after the French mathematician Pierre-Simon de Laplace, the Laplace operator is not just a mess of symbols, rather it is a reflection of symmetry. Mathematically, the Laplacian is a differential operator given by the divergence of the gradient of a function. Physically, it represents the flux density of the gradient flow of a function. The physical meaning of Laplace operator can be understood by a small example. Suppose, we have a certain chemical dissolved in some fluid. Then, the net rate at which a chemical dissolved in a fluid moving towards or away from some point is proportional to the Laplacian of the chemical concentration in the fluid at that point. However, in Poisson's equation, the Laplacian of Potential distribution in a system is proportional to the system's local charge distribution.

Poisson's Equation
Laplace Equation (1) & Laplacian in different Cordinate Systems
Courtesy: Slideserve.org

Origin of Poisson's Equation:

Mathematically, we can obtain the Poisson's equation by considering two things.. First: the Gauss law. And, Secondly: the fact that electric field can be written as a negative gradient of electric potential. The step wise step calculation and substitution of these two yields us the required equation.

Poisson's Equation
Derivation of Poisson's Equation
Courtesy: Slideserve.org , Slideshare.org

What if local charge distribution/charge density is Absent?

If the charge density in any region is zero, then the right hand side of our equation becomes zero. Here, The Poisson's equation becomes the Laplace equation. The Laplace equation has no source term, that is, no charge. This means that Laplace equation is homogeneous. Poisson's equation is just a generalized form of Laplace Equation. Just like every square is a rectangle but not every rectangle is a square. Similarly, all cases of Laplace equation are cases of Poisson's equation. But, not all the cases of the latter satisfy the Laplace Equation.

Importance of Poisson's Equation:

The Solutions of Poisson's equation help us to find an Electric Potential for a given charge distribution. Finding the potentials in this manner is one of the fundamental task for researchers. If the charge distribution follows a Boltzmann distribution, then this equation enhances to the Poisson–Boltzmann equation. This is a useful equation to understand different physiological interfaces, polymer science, electron interactions in a semiconductor, and many more. Apart from this, this equation also has many engineering applications. It also plays a major role in study of astrophysical plasmas and many more phenomenons in its several modified forms.

Also Read: What does Electrostatic Potential really means?

Authors message:

I hope this article gave you a basic idea about this powerful equation in Electrostatics. Equation are the fundamental tools to solve any theoretical problem. But, sometimes, understanding these equations itself becomes a problem. So, in our Month of Equation Series, we aim to bring some of the principal equation in a manner easy to comprehend. I hope you had a smooth read!

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