The meaning of the delta dirac function

[anban]

Homework Statement

For a function ρ(x,y,z) = cδ(x-a), give the meaning of the situation and describe each variable.

Homework Equations

As far as units go, I know that:

ρ(x,y,z) = charge density = C/ m^3
δ(x-a) = 1/m
and if those two are correct, then b must have units of (C/m^2), which is some sort of surface charge distribution.

The Attempt at a Solution

I'm new to delta dirac functions and my understanding is very superficial. When the function is not inside an integral, I especially don't understand it!

I think that "a" represents some sort of center point where there is infinite charge density, and x is the distance from the center.

Any guidance would be greatly appreciated.

 

[tiny-tim]

hi anban!

ρ(x,y,z) = cδ(x-a)
…
I think that "a" represents some sort of center point where there is infinite charge density, and x is the distance from the center.

no, x is one of the three coordinates, (x,y,z)

hint: describe where is δ(x-a) ≠ 0 ? ​

 

[anban]

I'm not sure that I understand-- does δ(x-a) ≠ 0 when x=a?

The way I am thinking about this now is that x-a is some coordinate point along a line?

 

[tiny-tim]

eg hi anban!

(just got up :zzz:)

The way I am thinking about this now is that x-a is some coordinate point along a line?

your mathematical language is rather strange

x-a is just an expression, you need to put it in a sentence

eg x-a = 0 is the whole plane x = a,

ie the points (a,y,z) for any values of y and z ​

 

[Nickie]

The delta dirac function, also known as the Dirac delta function, is a mathematical tool used to model a point charge or point mass in physics. In this case, the function ρ(x,y,z) = cδ(x-a) represents a charge density at a specific point, a, in space. The variable c represents the magnitude of the charge at that point, while x, y, and z represent the coordinates in a three-dimensional space.

As you correctly stated, the units of ρ(x,y,z) are C/m^3, which represents the amount of charge per unit volume at the point a. The delta dirac function is often used in conjunction with the integral of a function, where it acts as a weight or concentration factor at a specific point.

In this case, the function ρ(x,y,z) = cδ(x-a) can be interpreted as a charge density distribution on a flat surface, where x represents the distance from the center point a. The units of ρ(x,y,z) would then be C/m^2, representing the amount of charge per unit area on the surface.

It is important to note that the delta dirac function is not a regular function, as it is not defined at x = a and is only non-zero at that specific point. This makes it a useful tool for modeling point charges or point masses in physics, where the charge or mass is concentrated at a single point.