Finding Total Charge in Sphere w/ δ(r) Function

In summary, to find the total charge in a sphere with a given radius R and charge density function δ(r), use the formula Q = ∫δ(r)dr over 0 to R, taking into account the change of variables in spherical coordinates.
  • #1
iejedijuerijr
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Homework Statement



I was given a sphere with radius R and a charge density function δ(r),where r is the distance from the center of the sphere. I want to find the total charge in the sphere.


Homework Equations



I know that Q = δV


The Attempt at a Solution



I tried integrating V ∫δ(r)dr over 0 to R, but I'm not sure that will give me the total charge, but am unsure that's the right way to go about this. is it? thanks.
 
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  • #2
I believe your approach is correct.

Just remember that when integrating from 0 to R in spherical coordinates your must remember to use the factors associated with this change of variable.

That is, I think you will want to integrate:

Q = [itex]\int^{2\pi}_{0}[/itex] [itex]\int^{\pi}_{0}[/itex] [itex]\int^{R}_{0}[/itex] [itex]\delta[/itex](r) r[itex]^{2}[/itex] sin[itex]\theta[/itex] dr d[itex]\theta[/itex] d[itex]\phi[/itex]

where Q is the total charge enclosed by the sphere and R is the radius of the sphere.
 

What is a δ(r) function and how does it relate to finding total charge in a sphere?

A δ(r) function, also known as a Dirac delta function, is a mathematical function used to represent an infinitely narrow and infinitely tall spike at the origin. In the context of finding total charge in a sphere, the δ(r) function is used to represent the charge density at a specific point in the sphere.

How do you calculate total charge in a sphere using the δ(r) function?

To calculate total charge in a sphere using the δ(r) function, you would integrate the δ(r) function with respect to volume. This means you would multiply the charge density, represented by δ(r), by the volume element dV and then integrate over the entire volume of the sphere. This will give you the total charge in the sphere.

What are the limitations of using the δ(r) function to find total charge in a sphere?

The δ(r) function is a mathematical idealization and is not a physically realizable function. Its use in finding total charge in a sphere assumes that the charge density is concentrated at a single point, which is not always the case in real-world situations. Additionally, the δ(r) function cannot accurately represent continuous charge distributions.

Can the δ(r) function be used to find total charge in other shapes besides a sphere?

Yes, the δ(r) function can be used to find total charge in any three-dimensional shape, as long as the shape has a defined volume. However, the shape must have a defined origin point, as the δ(r) function is centered at the origin.

How does the δ(r) function relate to the concept of charge density?

The δ(r) function is used to represent the charge density at a specific point in a three-dimensional space. It allows for the calculation of total charge by integrating the charge density over the entire volume of the object. In this way, the δ(r) function is a useful tool in understanding the distribution of charge within a given object.

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