# Exploring Dirac Delta Function: Using to Express 3D Charge Distributions

• ZetaX
In summary, the conversation discusses the use and application of the Dirac delta function in expressing charge distributions in spherical coordinates. The main question is about integrating the delta function with the appropriate limits to get the correct result. There is also a reminder to understand what each term in the notation represents and to use the correct units.
ZetaX
Hello community, this is my first post and i start with a question about the famous dirac delta function.
I have some question of the use and application of the dirac delta function.
My first question is:
Using Dirac delta functions in the appropriate coordinates, express the following charge distributions as three-dimensional charge densities [PLAIN]http://www.physics.sfsu.edu/%7Edsoto/704/1.3_files/image002.gif.[/I] [PLAIN]http://www.physics.sfsu.edu/%7Edsoto/704/1.3_files/image003.gif[B][I](a)[/I][/B][I] In spherical coordinates, a charge [PLAIN]http://www.physics.sfsu.edu/%7Edsoto/704/1.3_files/image005.gifuniformly distributed over a spherical shell of radius R.So, i know the definition of dirac delta function in spherical coordinates ist $$\rho(\vec{r})=\sum_{k=0}^{\infty}\frac{1}{r_k^2\sin\theta}\delta(r-r_k')\delta(\theta-\theta')\delta(\phi-\phi')$$.

Moreover i know $$Q=\int\rho(\vec{r})d^3r$$. So i substitute the density function into the integral and transform it into spherical coordinates.

$$Q=\int\int\int\sum_{k=0}^{\infty}\frac{1}{r_k^2\sin\theta}\delta(r-r_k')\delta(\theta-\theta')\delta(\phi-\phi')r^2\sin\theta drd\theta d\phi$$
Now i change the sum into the integral and the limits become:

$$\int_0^{2\pi}\int_0^{\pi}\int_{-\infty}^{\infty}\delta(r-R)drd\theta d\phi$$
So here is the problem, if i integrate i get a incorrect answer...

What is my mistake?

Greetings! :)

Last edited by a moderator:
Shouldn't be your limits of integration for ##r## would be from ##0## to ##R##?

i.e. does it make sense for the radius to have a negative value?
Make sure you understand what each term in the notation is for.

What does the integral need to look like to come out right?
You certainly don't have the right units - you seem to have used ##\rho(\vec r)## in the first two equations to mean different things. In the first, it is the dirac delta function in spherical coordinates and in the second it is a charge density ... then you treat the dirac delta function as a charge density which it is not.

Last edited:

## 1. What is the Dirac Delta function and how is it used in expressing 3D charge distributions?

The Dirac Delta function is a mathematical concept that represents an infinitely tall and thin spike centered at the origin, with an area of 1 under its curve. In physics, it is used to model point-like particles or charges by placing them at the origin and assigning them a value of 1. This allows for the expression of 3D charge distributions in a simpler and more compact manner.

## 2. How does the Dirac Delta function relate to the concept of a point charge?

The Dirac Delta function is often used to represent point charges in physics. This is because the function has a value of 1 at the origin and a value of 0 everywhere else, mimicking the behavior of a point charge. By multiplying the function with the charge value, we can express the charge distribution at a point in space.

## 3. Can the Dirac Delta function be used to represent continuous charge distributions?

Yes, the Dirac Delta function can be used to represent continuous charge distributions. This is done by integrating the function over the volume of the distribution. The result is a continuous charge density function that can be used in calculations involving electric fields and potentials.

## 4. How does the use of the Dirac Delta function simplify calculations involving 3D charge distributions?

The use of the Dirac Delta function allows for a significant simplification of calculations involving 3D charge distributions. By representing the charges as point-like particles, we can use the superposition principle to sum the contributions of each charge, making the calculations much more manageable and efficient.

## 5. Are there any limitations to the use of the Dirac Delta function in expressing 3D charge distributions?

While the Dirac Delta function is a useful mathematical tool, it does have some limitations in representing 3D charge distributions. For example, it cannot be used to model continuous charge distributions with finite size or shape. It also assumes that the charges are point-like particles, which may not always be accurate in real-world scenarios.

Replies
25
Views
2K
Replies
5
Views
3K
Replies
2
Views
1K
Replies
13
Views
3K
Replies
1
Views
2K
Replies
3
Views
2K
Replies
7
Views
2K
Replies
2
Views
4K
Replies
1
Views
1K
Replies
6
Views
2K