# The meaning of the delta dirac function

• anban
In summary, ρ(x,y,z) = cδ(x-a) represents a function with units of charge density (C/m^3) where x is one of the three coordinates and "a" represents a point along the x-axis with infinite charge density. The expression x-a = 0 defines the whole plane x = a, meaning that any point (a,y,z) has an undefined charge density.
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.

hi anban!
anban said:
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 ?

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?

eg hi anban!

(just got up :zzz:)
anban said:
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

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.

## What is the delta dirac function?

The delta dirac function, also known as the Dirac delta function or impulse function, is a mathematical function that has a value of zero everywhere except at one point, where it has an infinite value. It is often used in physics and engineering to model point-like sources or idealized point masses.

## What is the purpose of the delta dirac function?

The delta dirac function is used to represent an impulse or instantaneous point of force or mass. It allows for a simpler mathematical representation of these phenomena, which can be useful in certain applications such as signal processing and modeling of physical systems.

## How is the delta dirac function defined?

The delta dirac function is defined as a limit of a sequence of functions, such as a Gaussian function, as their width approaches zero and their amplitude approaches infinity. It is often represented by the symbol δ(x) or δx and is defined as zero for all values of x except at x=0 where it is infinite.

## What are some properties of the delta dirac function?

The delta dirac function has several important properties, including: it is an even function, it is an odd function multiplied by infinity, its integral over all real values is equal to 1, and it has a sifting property where it picks out the value of a function at the point where it is multiplied by the delta dirac function.

## How is the delta dirac function used in applications?

The delta dirac function has many applications in physics, engineering, and mathematics, including: in signal processing to model impulse signals, in electrical engineering to represent idealized point charges or currents, in quantum mechanics to represent point-like particles, and in differential equations to represent boundary conditions. It is also used as a tool for simplifying mathematical calculations in various fields.

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