Gauss' Law: Calculating charge enclosed

In summary, the conversation discusses the differences in calculating the charge enclosed by a nonconducting sphere in two different situations: one with uniformly distributed charge and one with a charge per unit volume given by a function. In the first situation, the charge density is uniform throughout the sphere, making it easy to calculate the total charge by multiplying the charge density by the volume of the sphere. In the second situation, the charge is more densely concentrated on the outermost parts of the sphere and less densely concentrated on the center, requiring the use of integration to sum the total charge enclosed on each infinitesimal shell within the sphere.
  • #1
henry3369
194
0

Homework Statement


I've been doing a few Gauss' Law problems and I'm slightly confused about calculating charge enclosed by a nonconducting sphere.

So I have done 2 problems that involve finding the electric field inside nonconducting spheres:
1. Charge is uniformly distributed
2. Charge per unit volume given by the function ρ(r) = pr, where p is some constant.

I know how to solve both, but I'm not sure why I have to take different approaches to finding charge enclosed.

Homework Equations


EA = Qenc

The Attempt at a Solution


For (1), I was able to calculate Qenclosed using the fact that the charge density is Q/V, where V is the volume of the entire sphere, then multiplying it by the volume of the inner sphere, dV. Then I rearranged for E and got kQr/R3.

My question is, why can't I multiply the the volume of the inner sphere by the function ρ(r), to find the charge enclosed? Instead, I have to integrate ρ(r)*dV = ρ(r)*(4πr2) from 0 to r.

Both Q/V (1) and ρ(r) (2) provide the charge per unit volume, I don't understand why finding Qenclosed differs between the two.
 
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  • #2
In the first situation, the charge per unit area in each "shell" within the sphere is precisely the same as every other shell, because the charge density is uniform throughout. Since you know the charge density is uniform, if you know the volume of the sphere it's a very simple matter to get the total charge by multiplying the charge density by the volume of the sphere.

In the second situation the charge isn't uniformly distributed -- it's more like a gradient of charge. The charge is more densely concentrated on the outermost parts of the sphere and less densely concentrated on the center, so it's not as simple as considering the charge density to be some constant you can just multiply by the volume. The charge per unit area at each "shell" an infinitesimal width "dA" within the sphere, however, is constant, so you can multiply the charge per unit area by the area of the shell to find the charge contained in each infinitesimal shell. I hope that's fairly clear without the use of diagrams.

Knowing that, you'd need to sum the total charge enclosed on the area of each thin shell within the sphere to find the total charge enclosed within the volume of your sphere -- which requires you to use integration.

That's the difference, I hope it helps.
 

What is Gauss' Law?

Gauss' Law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the charge enclosed within that surface. In simpler terms, it explains how electric charges are distributed and how they affect the surrounding electric field.

How is Gauss' Law applied to calculate charge enclosed?

To calculate the charge enclosed within a closed surface using Gauss' Law, we use the equation Q = ε0ΦE, where Q is the enclosed charge, ε0 is the permittivity of free space, and ΦE is the electric flux through the surface. This equation can be rearranged to solve for Q, which is the desired charge enclosed.

What are the units of charge used in Gauss' Law?

The units of charge used in Gauss' Law are typically Coulombs (C) or Coulombs per meter squared (C/m2). These units represent the amount of electric charge present within a given area or volume.

What are some real-life applications of Gauss' Law?

Gauss' Law has many practical applications, including calculating the electric field inside a charged spherical or cylindrical conductor, determining the electric field between parallel plates, and analyzing the behavior of charged particles in electric fields. It is also used in engineering and physics to design and understand various electrical systems and devices.

What are the limitations of Gauss' Law?

Gauss' Law is based on the assumption that the electric field is uniform and symmetric around a charge distribution, which may not always be true in real-life situations. It also does not take into account the effects of magnetic fields, which can be important in certain scenarios. Additionally, Gauss' Law is only applicable to static electric fields and cannot be applied to dynamic systems.

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