Determining total charge on the surfaces of spherical conductors

[RichardEpic]

In the attached picture is all of the information to complete this problem. The picture is of a solid sphere at the center of a hallow sphere, both of which are conductors.

The question asks to find the total charge of the exterior and interior surfaces of the hollow conductor, as well as the exterior surface of the solid conductor...

I've attempted using Gauss' Law in various ways, but nothing seems to work, and I'm unsure of how to calculate the net E-field to 15,000 N/C, as indicated in the picture. If someone could point me in the right direction it's be greatly appreciated! :]

 

[WJSwanson]

Gah. Just as I was about to head to bed. I can't get into much detail until the morning, but the advice I can give is to remember your equation for the field due to a spherical charge distribution as well as 2 important rules:

1. In a spherical distribution the excess charges on a conductor align uniformly on the outer surface.
2. The charges will be aligned such that there is no net electric field inside the conductor itself (although there may be a net field in the gap between the inner and outer conductors if there is a net charge on the inner conductor).

Choosing Gaussian surfaces to take advantage of these properties, you should be able to determine the signs (do this first) and magnitudes of the net charges at play. Hope this helps. If you're still unclear on where to go, I'll return to this thread in the morning.

 

[RichardEpic]

Hmm...then it would seem to me that the point outside both the spheres would be easiest to calculate! By what you've said...do you mean that the inner surface of the Hallow sphere is actually inducing no E-field at that point..?

I think I am only confused now about the Inner E-field, I will attempt the Gaussian surfaces at these points and see how they go...thanks!

 

[RichardEpic]

Wow...I got it! The inner charge has to be opposite the inner charge, and I was able to solve for the two points by utilizing gaussian surfaces to give me the charges of the exterior surface of both! Your help is greatly appreciated!

WOO!

 

[rwisz]

To determine the total charge on the surfaces of the spherical conductors, we can use the Gauss' Law equation: Φ = Q/ε0, where Φ is the electric flux through a closed surface, Q is the total charge enclosed by that surface, and ε0 is the permittivity of free space.

First, let's consider the hollow conductor. Since the electric field is uniform and perpendicular to the surface of a conductor, the electric flux through the surface of the hollow conductor will be equal to the electric field multiplied by the surface area of the hollow conductor (Φ = E * A). We are given that the electric field is 15,000 N/C, and the surface area of the hollow conductor can be calculated using the formula for the surface area of a sphere (A = 4πr^2).

Next, we need to determine the total charge enclosed by the surface of the hollow conductor. Since the hollow conductor is a closed surface, the total charge enclosed will be equal to the net charge on the surface. This can be calculated by multiplying the surface charge density (σ) by the surface area (A). We are given that the surface charge density is -1.5 μC/m^2, so we can calculate the total charge on the surface of the hollow conductor.

Using the equation Φ = Q/ε0, we can now solve for Q and determine the total charge on the exterior and interior surfaces of the hollow conductor.

For the solid conductor, we can use the same approach. The only difference is that the electric field inside a solid conductor is zero, so the electric flux through the surface of the solid conductor will be zero. Therefore, the total charge on the exterior surface of the solid conductor will be equal to the total charge on the interior surface of the hollow conductor.

By following these steps, we can determine the total charge on all surfaces of the spherical conductors. It is important to note that the direction of the electric field inside the hollow conductor will be opposite to that of the electric field outside the solid conductor, due to the charges on the surfaces. This is why we use the negative value for the surface charge density on the hollow conductor.

I hope this helps guide you in the right direction. Keep in mind that Gauss' Law is a powerful tool for determining the electric field and charge distribution in various scenarios, so it is important to practice using it in different situations to become more familiar with its applications.