Numerical modelling of electrostatic potential energy of a system

In summary, you have a good understanding of the relationship between Coulomb force, electric field, electric potential, and electrostatic potential energy. However, your method for numerically computing the total energy of an electric field needs to consider the entire charge distribution and use appropriate numerical methods.
  • #1
chawk
4
0
Hi there,

Lately, I've been trying to gain a deeper understanding of the relationship between the Coulomb force F, electric field E, electric potential V, and electrostatic potential energy U.

In regards to electrostatic potential energy, as I understand it, if you had, say (made up a few values):

stationary point charge Q:

[tex]Q = 1.9258\cdot10^{-5} C[/tex]

you bring in another charge q:

[tex]q = 1.9258\cdot10^{-5} C[/tex]

to a distance of 66.6cm and hold it, the electrostatic potential energy of this 2 charge system is:

[tex]U = \frac{Qq}{4\pi\varepsilon_0r}[/tex]
[tex]U = \frac{(1.9258\cdot10^{-5})^2}{4\pi\varepsilon_0(0.666)} = 5 J[/tex]

If I then released q, it would gain 5 J of kinetic energy and move with a velocity calculable from [tex]\frac{1}{2}mv^2[/tex] (relativistic corrections made as needed).

Is this overall idea correct?

Then, in learning about capacitance and electric field energy, I come across the equation for the total energy of an electric field:

[tex]U = \int_{V}{\frac{1}{2}\varepsilon_0|E|^2dV}[/tex]

I was wondering if I could numerically compute the 5 J total system energy using this equation somehow. My initial thought was stepping over a uniform volume (a cube) around the system, slicing it into tiny chunks which serve as my differential volume elements, and calculating the vector sum of the individual E field contributions from the 2 charges:

[tex]E_{net} = E_Q + E_q[/tex] and then a "differential energy" I guess: [tex]dU = \frac{1}{2}\varepsilon_0|E|^2dV[/tex] where dV = (dimension of a slice)3. The dU's are calculated inside a triple nested for-loop, one for each coordinate, and totaled. My hope was that total would be approximately 5.0 J, but I soon realized something was conceptually quite wrong with this method.

For one, if I simulate and compute the field energy of a system with just the original stationary charge Q, the square of the field amplitude is positive everywhere and my total energy can reach enormous values. According to textbooks I'm reading, an isolated charge does not have electric field energy since it took no work to bring that charge to its location. This is an intuitive idea since the Coulomb force acts between two charges, but I can't seem to wrap my head around an intuitive way to discretize the volume around a system in order to numerically compute the results obtained analytically.

Perhaps I have some fundamental misunderstanding of the field energy equation, or maybe I'm just too tired :confused:

Anyone have any pointers for this one?
 
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  • #2



Hello,

It seems like you have a good grasp on the concept of electrostatic potential energy and its relationship to Coulomb force, electric field, and electric potential. Your example of calculating the electrostatic potential energy for a two-charge system is correct, and your understanding of the release of potential energy leading to kinetic energy is also accurate.

As for your second question about numerically computing the total energy of an electric field using the equation U = \int_{V}{\frac{1}{2}\varepsilon_0|E|^2dV}, there are a few things to consider.

Firstly, the equation you are using is for the total energy of an electric field in a given volume, not just for a single charge. So when you simulate and compute the field energy of a system with just the original stationary charge Q, the enormous values you are getting are because the volume you are considering is infinite. In reality, electric fields exist in a finite space, and the total energy of the field is finite as well.

Secondly, the equation you are using assumes a continuous distribution of charges, not just point charges. So when you try to discretize the volume around a system, you are not taking into account the actual distribution of charges and their corresponding electric fields. This is why your results do not match the analytical solution.

In order to accurately compute the total energy of an electric field using this equation, you would need to consider the entire distribution of charges and their corresponding electric fields within the given volume. This can be a complex calculation, but it can be simplified by using numerical methods such as finite element analysis.

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

1. What is numerical modelling of electrostatic potential energy?

Numerical modelling of electrostatic potential energy involves using mathematical equations and computer algorithms to simulate and predict the distribution of electric charge and resulting potential energy within a system.

2. What are the applications of numerical modelling of electrostatic potential energy?

This technique is commonly used in fields such as physics, chemistry, and engineering to study the behavior of charged particles and understand how they interact with each other and their surroundings.

3. How is electrostatic potential energy calculated in numerical modelling?

The electrostatic potential energy of a system is calculated by summing up the potential energy contributions of each individual charge, taking into account their distances and charges as described by Coulomb's law.

4. What are the limitations of numerical modelling of electrostatic potential energy?

One limitation is that it assumes a static system, meaning that the charges do not move or change position. It also relies on simplifying assumptions and may not accurately represent real-world scenarios.

5. How can numerical modelling of electrostatic potential energy be validated?

Validation can be done by comparing the results of the numerical model to experimental data or other established theoretical models. It is also important to test the sensitivity of the model to variations in parameters and conditions.

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