The Divergence of a Regularized Point Charge Electric Field

In summary, the problem involves calculating the divergence of a vector field representing the electric field of a point charge regularized by adding a constant in the denominator. The vector field is given by ##\left( \vec r \right) = \frac {\vec n} {(r^2+a^2)}## where ##\vec n = \frac {\vec r} r##. The solution involves calculating the partial derivatives of the vector field and integrating over all space to confirm that the divergence becomes proportional to the δ-function in the limit a -> 0.
  • #1
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1. Problem: Consider vector field A##\left( \vec r \right) = \frac {\vec n} {(r^2+a^2)}## representing the electric field of a point charge, however, regularized by adding a in the denominator. Here ##\vec n = \frac {\vec r} r##. Calculate the divergence of this vector field. Show that in the limit a -> 0 the divergence becomes proportional to the δ-function.

Homework Equations


∇⋅ = ## \frac \partial {\partial x} + \frac \partial {\partial y} + \frac \partial {\partial z}##

The Attempt at a Solution


So it seemed pretty straight forward to me, but I feel like there's something fundamental that I'm not seeing.

##\vec r = \left( x, y, z\right)##

##r = \sqrt {x^2 + y^2 + z^2}##

∇⋅A## \left( \vec r \right) = {\frac \partial {\partial x}} \frac x {\left( x^2 + y^2 + z^2\right)^{1/2} \left( x^2 + y^2 + z^2 + a^2 \right)} + {\frac \partial {\partial y}} \frac y {\left( x^2 + y^2 + z^2\right)^{1/2} \left( x^2 + y^2 + z^2 + a^2 \right)} + {\frac \partial {\partial z}} \frac z {\left( x^2 + y^2 + z^2\right)^{1/2} \left( x^2 + y^2 + z^2 + a^2 \right)} ##

I don't have any trouble with the computation, rather I feel like I didn't set this up correctly. Can anyone confirm if I'm moving in the right direction? Thanks!
 
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  • #2
What you wrote is completely correct. At first sight, calculating this looks like a lot of work, but once you realize how symmetrical the expression you wrote down is, you can easily conclude that it suffices to calculate one of the three derivatives.
 
  • #3
Awesome, guess I'll start plodding away. And I guess just to make sure, for the second part, all I would need to do is set a = 0, integrate over all space, and confirm it equals 1?
 

What is the divergence of a vector field?

The divergence of a vector field is a measure of the outward flow of a vector field at a specific point. It represents the amount of "source" or "sink" at that point, and is a scalar value.

How is the divergence of a vector field calculated?

The divergence of a vector field is calculated using the dot product of the vector field and the del operator (∇). This results in a scalar function that represents the divergence at each point in the vector field.

What does a positive or negative divergence value indicate?

A positive divergence value indicates an outward flow or "source" at a specific point in the vector field. A negative divergence value indicates an inward flow or "sink" at that point. A divergence value of 0 indicates no flow at that point.

What is the physical significance of the divergence of a vector field?

The divergence of a vector field has physical significance in fluid dynamics, electromagnetism, and other fields. It represents the rate of change of the density of a fluid or the flow of a field, and is an important tool in understanding the behavior of these systems.

How can the divergence of a vector field be used in practical applications?

The divergence of a vector field is used in a variety of practical applications, such as predicting the flow of fluids, analyzing the behavior of electromagnetic fields, and understanding the behavior of weather patterns. It is also used in computer graphics to simulate realistic fluid and smoke effects.

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