Self energy
Understanding Self Energy
Self energy is a concept in physics that refers to the energy required to assemble a charge distribution from an initial state where the constituent charges are infinitely separated to a final configuration. This concept is particularly relevant in electrostatics, where it is used to quantify the energy stored in an electric field created by a static charge distribution.
Definition and Formula
The self energy (U) of a charge distribution can be calculated by integrating the work done to bring each infinitesimal charge element (dq) from infinity to its final position within the charge distribution. Mathematically, it can be expressed as:
$$ U = \frac{1}{2} \int \rho(\vec{r}) \phi(\vec{r}) \, d^3r $$
where:
- $\rho(\vec{r})$ is the charge density at position $\vec{r}$.
- $\phi(\vec{r})$ is the electric potential at position $\vec{r}$ due to the charge distribution.
- The factor of $\frac{1}{2}$ is included to avoid double-counting the interaction energy between pairs of charge elements.
Examples
Example 1: Self Energy of a Point Charge
For a point charge $q$, the self energy is given by:
$$ U = \frac{1}{2} q \phi(\vec{r}) $$
Since the potential $\phi(\vec{r})$ due to a point charge is $\frac{q}{4\pi\epsilon_0 r}$, the self energy becomes infinite as $r \to 0$. This is a known issue in classical electrostatics and is addressed in quantum electrodynamics.
Example 2: Self Energy of a Uniformly Charged Sphere
For a uniformly charged sphere of radius $R$ and total charge $Q$, the self energy can be calculated by integrating the work done to assemble the sphere:
$$ U = \frac{3}{5} \frac{Q^2}{4\pi\epsilon_0 R} $$
This result is derived by considering the energy required to assemble the sphere shell by shell from the outside in.
Table of Differences and Important Points
| Property | Description | Relevance in Self Energy Calculation |
|---|---|---|
| Charge Distribution | The spatial arrangement of charges. | Determines the electric potential and thus the self energy. |
| Electric Potential | The work done per unit charge to move a test charge from infinity to a point in the field. | Integral part of the self energy formula. |
| Charge Density | The amount of charge per unit volume. | Used to calculate the infinitesimal work done to bring charge elements together. |
| Work Done | The energy required to move a charge in an electric field. | The self energy is the total work done to assemble the charge distribution. |
Calculation Steps
To calculate the self energy of a charge distribution, follow these steps:
- Determine the charge density $\rho(\vec{r})$ of the distribution.
- Calculate the electric potential $\phi(\vec{r})$ at every point due to the entire charge distribution.
- Integrate the product of charge density and electric potential over the entire volume of the charge distribution.
- Include the factor of $\frac{1}{2}$ to account for the self-interaction of the charge elements.
Conclusion
Self energy is an important concept in electrostatics and is essential for understanding the energy stored in electric fields. It is a measure of the work done to assemble a charge distribution and is influenced by the charge density, electric potential, and the spatial arrangement of charges. The calculation of self energy can be complex, depending on the geometry and distribution of charges, but it provides valuable insights into the properties of electric fields and the forces between charges.