Exploring Gauss' Law of Flux & Newton's Shell Method for Hollow Spheres

[physmeb]

While I do see how this makes sense using Newton’s Shell method, I don’t see how Gauss’ Law of Flux for a closed surface proves the same thing.

Both Gauss’ Law of Flux and Newton’s Shell method make perfect sense to me in showing that when dealing with a point outside the hollow conducting sphere, the sphere can be treated as a point charge equal to the sum of the surface charges on the hollow sphere (in Newton’s case it would be the sum of the masses). Since Gauss’ Law (as I understand it – maybe this is where my problem lies) is the sum of differentials of the component of the local E field normal to a differential area on the closed surface dA, multiplied by that differential area, yielding units of Newtons/Coulomb X metres squared; if applying this law to a spherical surface about a central charge Q, then the integral is simple, since the distance from any point on the surface to Q is R and we get that the flux for the sphere works out to be the Charge inside divided by epsilon zero. Fine makes, sense to me.

But in watching an MIT lecture the professor then applies this law to a hollow sphere and a point inside it. He draws a dashed closed surface inside the sphere upon which the point in question sits and says, well there’s no Q inside that imaginary control surface and thus there is no flux through the surface, i.e. no E field inside the surface. Voila. Voila? What in god’s name is he talking about? Is this some kind of reverse logic? I know there’s no charge inside the surface but there is charge outside the surface which could cause a flux through the surface but for overall field cancellation which Newton’s method actually does prove in a logic way. The MIT prof. professes implies that Guass’ law bypasses the need for Newton’s proof. I just don’t see how; I’d love to, but I just don’t see it. Your thoughts?


Ben.

 

[Yhaz]

An intuitive way to think of it if you are interested.

Think of an electric field line created by a charge entering the surface of a sphere or any surface which encloses some volume, is there a way for it to enter the surface and not leave it?

 

[maCrobo]

Gauss' Law let you calculate the flux through a surface without calculating the electric field at each point on the surface, it just cares about the charge inside the surface.
Imagine to have a spherical conductor and a point charge. In this case the electric field in between the sphere and the point charge (assuming the sphere has a certain charge on it) is created by both the electric field of the sphere and that of the point charge. Then you want to calculate the flux across a certain closed surface that contains the sphere. To do so you have two ways: you explicitly evaluate an expression for the electric field that allows you to calculate ∫EdA (and this E is given by the superimposition of the electric field given by the point charge with that given by the sphere) or you calculate the magnitude of the charge inside the surface so to do q/ε0. Both results represent the flux across our surface.
Take the case of a sphere with a hollow inside, it's clear that inside there is no charge. If you want to calculate the flux of the electric field inside the sphere you can take a surface inside the hollow and see the charges.

 

[SammyS]

Hello physmeb . Welcome to PF ! The way I look at this is as follows.

From the very nature of conductors, it should be clear that (under static conditions)):

1. The only place excess charge can reside is on a surface.

2. The electric field within conducting material is zero.​

So, whenever you draw a Gaussian surface completely with a conductor, the electric flux passing through that surface is zero, independent of Gauss's Law. Gauss's Law then comes into play and tells you that the net charge within that surface is zero. For instance; If there's a cavity within a conductor which contains an isolated charge, Q, then there is a charge of -Q distributed on the surface of that cavity.

I hope that helps.

 

[asteriatic]

Dear Ben,

Thank you for your inquiry about Gauss' Law of Flux and Newton's Shell method for hollow spheres. It is understandable that you may have some confusion about how these two principles are related and how they prove the same thing. Let me try to provide some clarification.

Firstly, you are correct in your understanding of Gauss' Law of Flux. It states that the electric flux through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of free space (epsilon zero). This law is based on the principle of superposition, which states that the electric field at a point due to multiple charges is equal to the sum of the individual electric fields at that point.

Now, let's look at how this applies to a hollow conducting sphere. As you mentioned, when dealing with a point outside the sphere, we can treat the sphere as a point charge located at its center. This is because the electric field at any point outside the sphere is the same as that of a point charge located at the center, due to the principle of superposition. Therefore, we can use Gauss' Law to calculate the electric flux through a closed surface surrounding the sphere, and it will be equal to the charge enclosed by that surface divided by epsilon zero.

However, when we consider a point inside the hollow sphere, things get a bit more complicated. The electric field inside the sphere is not the same as that of a point charge located at the center. This is because the charges on the inner surface of the sphere create an electric field that is opposite in direction to the field created by the charges on the outer surface. This results in a net electric field of zero inside the sphere.

So, when the MIT professor draws a dashed closed surface inside the sphere, he is essentially creating a Gaussian surface that encloses no charge. This means that according to Gauss' Law, the electric flux through this surface must also be zero. This is a direct consequence of the fact that the electric field inside the sphere is zero.

Now, you may be wondering how this relates to Newton's Shell method for hollow spheres. The key concept here is that of symmetry. Both Gauss' Law and Newton's Shell method rely on the symmetry of the problem to simplify the calculations. In the case of a hollow conducting sphere, the symmetry allows us to treat the sphere as a point charge at its center, and also allows us to use Gauss' Law to show that the electric field inside the sphere is