How Does a Coaxial Cable's Capacitance Change with Different Configurations?

[bdubb427]

Help please w/ the following questions

Homework Statement

1.
A coaxial cable is given with two concentrically placed dielectrics. What is the capacitance of this structure? If an ungrounded metal shell is placed at a certain distance from the inner electrode, what will be the capacitance now? Plot the magnitude of field and potential as a function of radius in both cases.

Homework Equations

C = Q / V = [2 * pi * epsilon * length] / [ln(b/a)] where b is the radius of the entire cable (outer and inner conductor) and a is the radius of just the inner conductor

Homework Statement

2. A charged spherical shell is placed 1m above a ground metal plate. Find electric field.

Homework Equations

besides coulomb's law, i don't have any other relevant equations for this.


Our textbook is really hard to understand, and the examples given in the text are even harder to follow. the book is Fundamentals of Applied Electromagnetics by Ulaby.. Any suggestions for another book are nice, but my exam is coming up and finding another book is just not an option at this point. Please help if u can. thank you...

 

[JoAuSc]

For problem 1, you have four layers: a charged conducting cylinder, an inner dielectric layer, an outer dielectric layer, and an outer conducting shell. Because you have two dielectrics, the equation you provided (with just one epsilon) won't work. You'll need to rederive that expression for two dielectrics. I believe this would require a.) finding the electric field in the presence of a dielectric, which might be as simple as switching epsilon_0 with epsilon_1 in region 1 and epsilon_2 in region 2 (however you decide to label them); b.) calculating the voltage differences between the boundaries from the electric fields; and c.) calculating Q/V. I am not sure if adding in a thin metal shell would make any difference.

The second question seems much easier. First, with the method of images we know that this problem is equivalent to one where we have two spheres with opposite charges which are 2m apart. Gauss's law gives you the E-field for each sphere in the absence of the other sphere, and by the superposition principle you just add their fields to get the total.

 

[baba89]

Hello,

Applied Electromagnetics is a very interesting and complex field of study, so it is understandable that you are struggling with the material. Let me try to provide some guidance and help with the questions you have provided.

1. The capacitance of a coaxial cable with two concentric dielectrics can be calculated using the equation you have provided, where b is the radius of the entire cable (outer and inner conductor) and a is the radius of just the inner conductor. However, keep in mind that the dielectric constant of each dielectric material will also affect the capacitance.

If an ungrounded metal shell is placed at a certain distance from the inner electrode, the capacitance will change. This is because the presence of the metal shell will act as a third conductor, altering the electric field between the inner and outer conductors. To calculate the new capacitance, you can use the same equation, but you will need to take into account the distance between the inner and outer conductors and the metal shell. You can also use the method of images to simplify the calculation.

To plot the magnitude of the electric field and potential as a function of radius, you can use the equations for electric field and potential in a coaxial cable, which can be found in your textbook. You can also use software such as MATLAB or Mathematica to plot these functions.

2. To find the electric field of a charged spherical shell placed above a ground metal plate, you can use Coulomb's law. However, keep in mind that the presence of the metal plate will also affect the electric field. You can use the method of images to simplify the calculation and take into account the presence of the metal plate.

I understand that your textbook may be difficult to understand, but it is important to work through the examples provided and try to understand the concepts. You can also use other resources such as online lectures or tutorials to supplement your learning. If you have specific questions or are struggling with a particular concept, it may be helpful to reach out to your professor or a tutor for assistance.

I hope this helps and good luck with your exam!