Conductors with off-centered cavity

In summary, the conversation discusses the application of Gauss's law to find the E-field lines outside a solid spherical conductor with a charge placed in an off-center spherical cavity. It is established that the E-field outside the conductor is not affected by the position of the cavity or the shape of the conductor. The use of symmetry arguments and uniqueness is mentioned as a way to prove this. It is also noted that the outer surface of the conductor will have an equipotential surface of radius R and any potential outside the sphere will satisfy this boundary condition.
  • #1
raging
7
0
Suppose we had an uncharged solid spherical conductor with radius R, and some spherical cavity of radius a that is located b units above R, where a+b<R. If we place a charge inside the cavity at the center, how would you quantitatively express the E-field lines outside of the conductor?
 
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  • #2
This is NOT a homework. Concentric cavities are too easy. I'm wondering if we can somehow apply uniqueness here to show that it actually doesn't matter where the cavity is...
 
  • #3
The E field is easy to find using Gauss's law . And it doesn't matter where the charge is inside the sphere or the shape of the cavity.
 
  • #4
cragar said:
The E field is easy to find using Gauss's law . And it doesn't matter where the charge is inside the sphere or the shape of the cavity.
Exactly. You can kind of reason it out in a way, by knowing that charges like to stay on the very outside of a conductor. Think about it, say you've got a positive charge inside that off-centre cavity. That charge is going to pull negative charges towards the inner surface of that cavity to balance it out. Then, to balance out those negative charges in the conductor, positive charges are going to show up on the outside of the sphere (regardless of where the cavity is), and they'll create an electric field outside the sphere exactly the same as if you had just a charged sphere.

You can use Gauss' Law to prove every part of that by setting up surfaces just outside the cavity and just outside the sphere and using the relation:

[tex]\Phi_E = \int\vec{E}\cdot d\vec{A} = \frac{q_{enclosed}}{\epsilon_0}[/tex]

Where you know [tex]\vec{E} = 0[/tex] inside the conductor.
 
  • #5
True. But to solve that surface integral simply one must apply a symmetry argument. How do you arrive about that symmetry? The induced charge will not be evenly distributed on the outside of the sphere, but is there a way to apply uniqueness to arguing that?
 
  • #6
The outer surface of the conductor will be an equipotential surface of radius R. An outside potential phi=Q/r satisfies this boundary condition. It is thus the unique potential outside the sphere.
It doesn't matter what goes on in cavities within the conduciting sphere.
 

What is a conductor with an off-centered cavity?

A conductor with an off-centered cavity is a type of material that has a hollow space or cavity within its structure, which is not located at the center of the material. This cavity can impact the properties and behavior of the conductor.

What are the applications of conductors with off-centered cavities?

Conductors with off-centered cavities have a wide range of applications in various fields such as electronics, telecommunications, and energy storage. They can be used as high-performance capacitors, sensors, and in energy harvesting devices.

How do conductors with off-centered cavities differ from regular conductors?

Conductors with off-centered cavities have a unique structure that can enhance their electrical, mechanical, and thermal properties. This is due to the presence of the cavity, which can affect the flow of electrons and the overall behavior of the material.

What is the significance of the location of the cavity in conductors?

The location of the cavity in conductors can greatly influence their properties. For example, a cavity located closer to the surface of the material can affect its surface conductivity, while a cavity located towards the center can affect the bulk conductivity of the material.

What are the challenges in studying and utilizing conductors with off-centered cavities?

One of the main challenges in studying and utilizing conductors with off-centered cavities is their complex structure, which can make it difficult to accurately model and understand their behavior. Additionally, the fabrication of these materials can also be challenging and require specialized techniques.

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