Calculating D field in a region between two charged surfaces

[MNightShamylan]

1. Question: There is a wire with charge, surrounded by a metal cylinder with opposite charge. There is a dielectric surrounding the wire, going out half way to the outer cylinder. Calculate the D field in the region between the cylinder and wire.

Homework Equations

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Gauss's law for displacement: integral D * da = Q(free enclosed)

The Attempt at a Solution

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It seems to me that there will be a piecewise solution, one for the gaussian surface inside the dielectric and one just outside it (but just inside the outer cylinder). If the surface area of the gaussian cylinder is 2pirL, then is the D field just Q/(2pirL)? It seems like this is just way too simple, and I feel like I'm missing something. I also don't see how this takes into account the gaussian surface inside the dielectric, which should be different. Any guidance would be appreciated/

 

[anathema]

Hello,

Thank you for your question. It appears that you are on the right track in using Gauss's law for displacement to solve this problem. However, there are a few things to consider in order to arrive at the correct solution.

Firstly, you are correct in thinking that there will be a piecewise solution, as the electric field and displacement field will behave differently in the different regions. In order to properly calculate the D field in the region between the cylinder and wire, you will need to consider the contribution from both the wire and the outer cylinder.

One important factor to consider is the presence of the dielectric material. This will affect the electric field and displacement field in the region between the wire and cylinder. In order to account for this, you will need to use the relative permittivity of the dielectric, denoted by ε_r, in your calculations. This factor will modify the D field in the region between the wire and cylinder, and will also affect the boundary conditions at the interface between the dielectric and the metal cylinder.

Another important point to consider is the direction of the D field. Remember that the D field is defined as the electric flux per unit area, and has units of C/m^2. Therefore, you will need to carefully consider the direction of the electric field and how it contributes to the D field in the region between the wire and cylinder.

Overall, your approach of using Gauss's law for displacement is a good starting point, but you will need to consider the effects of the dielectric and carefully consider the direction of the D field in order to arrive at the correct solution. I hope this helps guide you in the right direction. Good luck with your calculations!