How to Find the Potential Within a Grounded Conducting Hollow Sphere?

• m0nk3y
In summary, the problem involves a grounded conducting hollow sphere containing a coaxial ring with a known charge distribution. The goal is to find the potential within the sphere along the z-axis, using a Green's function for the sphere. Previous attempts using separation of variables and the law of cosines have not been successful.
m0nk3y

Homework Statement

A grounded conducting hollow sphere of radius a contains a ring of radius b and charge per unit length $$\Lambda$$. The ring is coaxil to the z axis, and with the sphere lies a distance d about the center of the sphere such that d^2 + b^2 < a^2. Find the potential within the sphere along the z-axis

d^2 + b^2 < a^2.

The Attempt at a Solution

Honestly I don't know where to start. Reading Griffiths chapter 3.3 I attempted using the separation of variables but had no idea had to proceed. So now I am attempting on using the law of cosines and concentrating on the part where the radius of the charged ring and the center of the sphere make a triangle. However, i doubt this is right. Any help to push me in the right direction to solving this problem is greatly appreciated!

Thanks

The ring inducts an equivalent charge on the sphere, the very problem is to compute the charge distribution. But the inducted charges negates themselves so it would be enough to compute the potential only for the ring.

Last edited:
michalll said:
The ring inducts an equivalent charge on the sphere, the very problem is to compute the charge distribution. But the inducted charges negates themselves so it would be enough to compute the potential only for the ring.

That is not correct. The potential due only to the ring is not equipotential at the surface.
This problem has to be done using a Green's function for the sphere.

1. What is a grounded conducting hollow sphere?

A grounded conducting hollow sphere is a spherical object made of a conductive material, such as metal, that is connected to the ground. This means that any excess charge on the sphere is able to flow freely to the ground, making the sphere electrically neutral.

2. How does a grounded conducting hollow sphere work?

A grounded conducting hollow sphere works by redistributing any excess charge on its surface to the ground. This is due to the principle of electrostatic induction, where the presence of the ground provides a path for the excess charge to flow to and neutralize the sphere.

3. What are some common uses of grounded conducting hollow spheres?

Grounded conducting hollow spheres are commonly used in electrostatic experiments and demonstrations, as well as in equipment such as Van de Graaff generators. They are also used in lightning rods to protect buildings from lightning strikes.

4. What is the difference between a grounded and ungrounded conducting hollow sphere?

A grounded conducting hollow sphere is connected to the ground, allowing any excess charge to flow to the ground and neutralize the sphere. An ungrounded conducting hollow sphere, on the other hand, is not connected to the ground and can hold a net charge on its surface.

5. Can a grounded conducting hollow sphere have a net charge?

No, a grounded conducting hollow sphere cannot have a net charge. Any excess charge on the surface of the sphere will be neutralized by the ground, leaving the sphere with a neutral overall charge.

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