Potential Energies of Two Charged Cylinders

In summary: That energy is what you need to calculate for the second part.In summary, the problem involves calculating the energy per unit length stored in a solid cylinder with radius a and uniform volume charge density ρ. The first part of the problem is relatively simple, finding the potential energy density per length of the inside of the cylinder. However, the second part involves finding the potential energy of the hollow cylinder, which is not zero due to the energy required to assemble all the charges on the surface of a cylinder of radius R in a region of space where there was nothing initially. This energy can be calculated using the given equation, which yields a result of ##\frac{\lambda^2}{4\pi\epsilon_0}\left(1/
  • #1
Matt Chu
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0

Homework Statement



Problem 1.24 (this is unimportant; it's just a different way of calculating the potential energy of a solid cylinder) gives one way of calculating the energy per unit length stored in a solid cylinder with radius a and uniform volume charge density ##\rho##. Calculate the energy here by using ##U = \frac{\epsilon_0}{2} \int_{entire \ surface} E^2 dv## to find the total energy per unit length stored in the electric field. Don’t forget to include the field inside the cylinder.

You will find that the energy is infinite, so instead calculate the energy relative to the configuration where all the charge is initially distributed uniformly over a hollow cylinder with large radius ##R##. (The field outside radius ##R## is the same in both configurations, so it can be ignored when calculating the relative energy.) In terms of the total charge ##\lambda## per unit length in the final cylinder, show that the energy per unit length can be written as ##\frac{\lambda^2}{4\pi\epsilon_0}\left(1/4+ln(R/a)\right)##

(It's important to note that the potential energy involved in this problem is NOT the potential energy of a particle in the field created by the charged cylinders, but the potential energy of the charged cylinders themselves.)

Homework Equations



##U = \frac{\epsilon_0}{2} \int_{entire \ surface} E^2 dv##

The Attempt at a Solution



The first part of the problem, involving solving for the energy of a solid cylinder, is pretty simple. For this, I got that the potential energy density per length of the inside of the cylinder being:

##U_{in}/h = \frac{\pi \rho^2 R^4}{16 \epsilon_0}##

The external potential energy of the cylinder goes to infinity, as the problem states, as you get:

## U_{out}/h = \left. \frac{\pi \rho^2 R^4}{4 \epsilon_0} \ln(r) \right|_R^\infty ##

The second part of the problem, though, is confusing to me. It would seem that the potential energy of the hollow cylinder is zero, because the electric field inside is zero. In addition, the problem says that you can "ignore" the field outside ##R##, but I'm not sure how exactly that can be possible if the total potential energy is the sum of the potential energies of the internal and external areas.

I thought that maybe I could calculate the potential energy inside the hollow cylinder by doing it piece by piece, i.e. with differential areas rather than with the equation given, but logically it should still be zero as well.
 
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  • #2
Matt Chu said:
It would seem that the potential energy of the hollow cylinder is zero, because the electric field inside is zero.
It would seem that way. However, it takes some energy to assemble all those charges on the surface of a cylinder of radius R in a region of space where there was nothing initially.
 

Related to Potential Energies of Two Charged Cylinders

1. What is the definition of potential energy in the context of two charged cylinders?

Potential energy is the stored energy that results from the position or configuration of two charged cylinders in an electric field. It is a measure of the work that would be required to bring the two cylinders together from an infinite distance apart.

2. How is the potential energy of two charged cylinders calculated?

The potential energy of two charged cylinders is calculated using the formula PE = (k * Q1 * Q2) / d, where k is the Coulomb's constant, Q1 and Q2 are the charges of the cylinders, and d is the distance between them.

3. What factors affect the potential energy of two charged cylinders?

The potential energy of two charged cylinders is affected by the magnitude of the charges on the cylinders, the distance between them, and the medium between the cylinders (such as air, water, or a vacuum).

4. Can the potential energy of two charged cylinders be negative?

Yes, the potential energy of two charged cylinders can be negative if the charges on the cylinders are of opposite sign. This indicates a stable configuration, as the two cylinders will be attracted to each other.

5. How does the potential energy of two charged cylinders change with distance?

The potential energy of two charged cylinders decreases as the distance between them increases. This is because the electric force between the cylinders decreases with distance, resulting in a decrease in the potential energy.

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