Deriving Electric Field Energy Density

[foxjwill]

Homework Statement

I found on Wikipedia the formula for energy density of an electric field in a vacuum to be

[tex]U = \frac{1}{2}\epsilon_0 E^2.[/tex]

I was wondering if someone could point me in the right direction to figure out how this was derived.

Homework Equations

The Attempt at a Solution

I was thinking maybe using Gauss's law?

 

[Shooting Star]

It starts with total energy = 1/2[Integral over volume(rho*phi*dV)], where rho is the volume charge density, phi the potential.

Then this has to be written as e0/2[Integral over volume(div E*phi*dV)]. After that you can break up the integrand into two parts.

You can try to do it from here, but it'd best if you look up a book.

 

[Moetasim]

Hello! Thank you for your question. The formula for energy density of an electric field in a vacuum can be derived using the following steps:

1. Start with the definition of electric potential energy: U = qV, where q is the charge and V is the electric potential.

2. Use the definition of electric potential: V = kq/r, where k is the Coulomb constant and r is the distance from the charge.

3. Substitute this into the equation for electric potential energy: U = q(kq/r).

4. Use Coulomb's law: F = kq1q2/r^2, where F is the force between two charges q1 and q2.

5. Substitute this into the equation for electric potential energy: U = qF/kq1q2.

6. Use the definition of electric field: E = F/q, where E is the electric field.

7. Substitute this into the equation for electric potential energy: U = qEq/kq1q2.

8. Rearrange the equation to isolate the electric field: E = kq1q2U/q^2.

9. Use the definition of electric field: E = q/4πε0r^2, where ε0 is the permittivity of free space.

10. Substitute this into the equation for electric potential energy: U = (4πε0r^2)q^2/(4πε0r^2)^2.

11. Simplify the equation: U = q^2/8πε0r^4.

12. Multiply both sides by the volume of the electric field: U = q^2/8πε0r^4 * V = q^2/8πε0r^4 * (4πr^3/3).

13. Simplify the equation: U = q^2/6ε0r.

14. Substitute the definition of electric field: E = q/4πε0r^2.

15. Substitute this into the equation for energy density: U = (1/2)ε0E^2.

This is how the formula for energy density of an electric field in a vacuum, U = (1/2)ε0E^2, can be derived. I hope this helps guide you in the right direction for your homework.