Dirac Delta Function (electrodynamics)

In summary, the Dirac delta function is a mathematical tool used as a sampling function and impulse input to a system. It is useful in situations where there would be a singularity in a result or calculation, and can be used to describe the charge distribution in a given space. However, it can also result in infinite self energy when applied to point charges. It is important to be careful with the dimensions of the delta function and to use it when asked for the charge distribution.
  • #1
mateomy
307
0
I'm having a hard time grasping when I should use this little "function". I'm using Griffith's Intro to Electrodynamics and either he doesn't touch on it enough or I'm missing the point. From what I think I understand I'm to use it when there would be a singularity in a result or calculation(?). Not sure about this. I've been looking around a lot online but can't find anything that explicitly states "You should use this here." Maybe I'll never find it. In any case, does anyone have any pointers?
 
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  • #2
The Dirac delta function δ(x) is used

as a sampling function: ∫f(x)δ(x-x0)dx over -∞ to +∞= f(x0)

and as an impulse input to a system. It's very useful if for example a rectangular pulse is the input to a system of time constants τ such that duration << all τ. Then the pulse may be approximated by a delta function with coefficient = duration * amplitude.

E.g. input pulse of 2V and 10us would be 2e-5δ(t).

In any case you have to be very careful with the dimensions of δ(x), which has the dimension of 1/x.

The Laplace transform easily handles a delta function since its transform is just 1. And the "impulse response" of a network is its output with the input = δ(t).
 
  • #3
mateomy said:
I'm having a hard time grasping when I should use this little "function". I'm using Griffith's Intro to Electrodynamics and either he doesn't touch on it enough or I'm missing the point. From what I think I understand I'm to use it when there would be a singularity in a result or calculation(?). Not sure about this. I've been looking around a lot online but can't find anything that explicitly states "You should use this here." Maybe I'll never find it. In any case, does anyone have any pointers?

I'm a student of the upper undergraduate level EM course (we use Jackson's textbook). My limited experience tells me to use the Dirac delta "function" which is in fact a non regular distribution, when I want to have a useful mathematical way to describe WHERE the charge is, in space. More precisely when someone asks you to get the "charge distribution".
Let's take an example. Say you have charge over the surface of a sphere. They ask you "give me the charge distribution". You could write "0 everywhere except on the surface of the sphere x^2+y^2+z^2=r^2". However this isn't very useful and not even accurate for a reason I'll explain.
Something that must always be true is that [itex]\int _{\Omega } \rho (\vec x ) dV =Q_{\text{inside } \Omega }[/itex]. Here rho is the charge density and is written in terms of for example the Dirac distribution (maybe with a combination of the Heaviside "function" which is a regular distribution). So not only rho gives you information about where the charge lies but it tells you exactly how it is distributed. If you had answered "0 everywhere except on the surface of the sphere x^2+y^2+z^2=r^2" you wouldn't get a mathematically useful formula and you don't have the expression for rho such that when integrated over a region, it gives the total charge enclosed.
So to answer your original question, I'd say "when they ask you the charge distribution".
 
  • #4
That's a good way of putting it, thanks a lot. I am slowly getting where I should place it, but its sort of a rote thing and I'm not comfortable doing it primarily off that principle.

Thanks again!
 
  • #5
The delta function is a mathematical construct. In one sense, the construct allows us to model the electron as a point charge. In another sense, it is also simply a helpful mathematical tool. For example if I want to model a disc of charge in the xy plane I simply put a delta function of d(z)*H(R-r)*Q/piR^2 as the charge distribution (H Heaviside step).

The trouble with giving an electron a delta function potential is that the electron has infinite self energy as we look smaller and smaller.
 
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  • #6
Mindscrape said:
For example if I want to model a disc of charge in the xy plane I simply put a delta function of d(z)*d(r-R)*Q/piR^2 as the charge distribution.
Are you sure it shouldn't be [itex]\rho (\vec x )=\frac{q }{\pi R^2r} \delta \left ( \theta - \frac{\pi }{2} \right ) \Theta (r -R)[/itex]? I had this problem to solve (see https://www.physicsforums.com/showthread.php?t=591217).
The trouble with giving an electron a delta function potential is that the electron has infinite self energy as we look smaller and smaller.
I didn't know this was due to the delta function but that is indeed troublesome.
 
  • #7
Sure, if you want the disc in spherical coordinates. But that's slightly less intuitive so I went with cylindrical. :) (Oops, I actually did mess up the R though as I unwittingly made an annulus and just fixed it)

The infinite self energy is more of a function of the particle being a point charge, but since a delta function models a point charge I think you could attribute it to the delta. Since V=e/R when R->0 then U->infty. The trick to this problem is to just leave it alone and let QM fix it. :)
 
  • #8
Mindscrape said:
Sure, if you want the disc in spherical coordinates. But that's slightly less intuitive so I went with cylindrical. :) (Oops, I actually did mess up the R though as I unwittingly made an annulus and just fixed it)
Okay :)
The infinite self energy is more of a function of the particle being a point charge, but since a delta function models a point charge I think you could attribute it to the delta. Since V=e/R when R->0 then U->infty. The trick to this problem is to just leave it alone and let QM fix it. :)

I see. If I remember well my prof made a comment that if we calculate the energy via [itex]\frac{1}{8\pi} \int _{R^3 } \vec E \cdot \vec E dV[/itex], for a point charge, this diverges.
Intuitively I understand it as the energy required to form a charged sphere of radius r and taking the limit as r tends to 0. It becomes extremely hard to make it smaller and smaller due to repulsion. Eventually when you "reaches" the limit r=0, it has required infinitely energy to do so. Not sure this is really accurate though.
 

What is the Dirac Delta Function and why is it important in electrodynamics?

The Dirac Delta Function, denoted by δ(x), is a mathematical function that is used to model the behavior of point particles in quantum mechanics. In electrodynamics, it is important because it allows us to describe the electric and magnetic fields of point charges and point dipoles.

How is the Dirac Delta Function related to the electric charge density of a point particle?

The Dirac Delta Function is directly related to the electric charge density of a point particle. The charge density, ρ(x), at a point x is given by ρ(x) = qδ(x), where q is the charge of the particle. This means that the electric field at a point x due to a point particle is given by E(x) = q/4πε0δ(x)/x2, where ε0 is the permittivity of free space.

Can the Dirac Delta Function be used to calculate the potential and electric field of a continuous charge distribution?

Yes, the Dirac Delta Function can be used to calculate the potential and electric field of a continuous charge distribution. This is done by using the Dirac Delta Function to represent the charge density at each point in the distribution, and then integrating over the entire distribution to find the total potential and electric field.

How does the Dirac Delta Function behave under integration and differentiation?

The Dirac Delta Function has the useful property that it is zero everywhere except at x=0, where it is infinite. As a result, it integrates to 1 and differentiates to 0 at x=0, and is 0 everywhere else. This allows us to simplify calculations involving the Dirac Delta Function.

What are some practical applications of the Dirac Delta Function in electrodynamics?

The Dirac Delta Function has many practical applications in electrodynamics. It is used to model point charges and point dipoles, calculate the electric field of a continuous charge distribution, and solve boundary value problems in electrostatics and magnetostatics. It is also used in the study of electromagnetic waves and quantum mechanics.

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