Does the Presence of Conductors Affect Interface Dielectrics in a Coaxial Cable?

[jesuslovesu]

Homework Statement

A coaxial cable has a line charge running through it's center with a line charge density $$\lambda$$. There is a dielectric filled between the line charge and the first conducting shell cylinder and another dielectric between the outer cylindrical shell and the inner cylindrical shell.

Homework Equations

$$D_{1n} - D_{2n} = \rho_s$$

The Attempt at a Solution

My question is, in the equation to the bottom will rho have a value? Since the line charge is the only thing with direct charge on it, can I ignore the fact that there is a conductor separating the dielectrics?
I know the inner conductor will have a negative charge to counteract the field coming from the line charge, but with that equation isn't it applicable if the surface is directly charged?

so can i just say $$D_{1n} = D_{2n}$$ in this case anyway?

 

[anathema]

Hi there,

In this case, the line charge density \lambda will have a value since it is the source of the electric field in the coaxial cable. The equation D_{1n} - D_{2n} = \rho_s is applicable in this scenario because the surface charge density \rho_s is created by the electric field from the line charge. Therefore, you cannot ignore the presence of the conductor separating the dielectrics.

To answer your question, no, you cannot say D_{1n} = D_{2n} in this case. This is because the electric field between the dielectrics will be affected by the presence of the conductor and will not be equal on both sides. The equation D_{1n} - D_{2n} = \rho_s takes into account the effects of the conductor and the dielectrics.

I hope this helps clarify your question. Let me know if you have any further questions.

 

[Moetasim]

I can provide some clarification on the concept of interface dielectrics in this scenario.

Firstly, the equation you have mentioned is known as Gauss's law in differential form and is applicable to the case of a surface with a direct charge on it. In this case, the line charge in the center of the coaxial cable is the only source of direct charge, and the conducting shells act as a barrier between the line charge and the dielectrics.

When it comes to interface dielectrics, the equation you have mentioned may not be directly applicable. Instead, we can use the concept of boundary conditions to understand the behavior of the electric field at the interface between two dielectrics. These boundary conditions take into account the permittivity of the dielectrics and the direction of the electric field at the interface.

In this scenario, the electric field at the interface between the first dielectric and the inner conducting shell will be affected by the permittivity of the first dielectric, while the electric field at the interface between the two dielectrics will be affected by both the permittivity of the first and second dielectrics. Therefore, the equation you have mentioned may not be directly applicable, and we may need to use boundary conditions to analyze the electric field at the interfaces.

In conclusion, the presence of the conducting shells and the dielectrics will affect the electric field in the coaxial cable, and we cannot simply ignore their presence in the equation. As a scientist, it is important to consider all factors and use appropriate equations and principles to accurately analyze the situation.