1. Applied Electricity and Magnetism
ECE 3318
Applied Electricity and Magnetism
Spring 2017
Prof. David R. Jackson
ECE Dept.
Notes 26

2. Stored Energy
Goal:
Calculate the amount of stored electric energy in space, due to the presence of an electric field.
(Recall that the electric field is produced by a charge density.)

3. Stored Energy (cont.) The charge formulas require Charge formulas:
(volume charge density)
Note:
We don’t have a line-charge version of the charge formulas, since a line charge would have an infinite stored energy (as would a point charge).
(surface charge density)
The charge formulas require
Electric-field formula:
Note:
The charge and electric field formulas will give the same answer, provided that the region V includes all places where there is a charge or a field.
Please see the textbook for a derivation of these formulas.

4. We assume an ideal parallel-plate capacitor (no fields outside).
Example
Find the stored energy. + -
We assume an ideal parallel-plate capacitor (no fields outside).
Method #1
Use the electric field formula: so

5. Example (cont.) Hence We can also write this as Recall that
Therefore, we have

6. Example (cont.) Method #2 Use charge formula:
Use charge formula:
(We use the surface-charge form of the formula.)
We then have

7. Example (cont.) Also, we can write this as so or
Note: This same derivation (method 2) actually holds for any shape capacitor.

8. Alternative Capacitance Formula
Using the result from the previous example, we can write
where
UE = stored energy in capacitor
V = voltage on capacitor
This gives us an alternative method for calculating capacitance.

9. Alternative Capacitance Formula (cont.)
Find C (using the alternative formula).
Ideal parallel plate capacitor
Assume:

10. Alternative Capacitance Formula (cont.)
Therefore so

11. Solid sphere of uniform volume charge density (not a capacitor!)
Example
Find the stored energy.
Solid sphere of uniform volume charge density (not a capacitor!)
Gauss’s Law:
r < a
r > a

12. Example (cont.)
Hence we have

13. Example (cont.) Result: Denote so that We then have
This is the energy that it takes to assemble (force together) the charges inside the cloud from infinity.
Denote
so that
We then have

14. Example (cont.) Final result: Note:
It takes an infinite amount of energy to make an ideal point charge!

15. This is one form of the “classical electron radius.”
Example
Find the “radius” of an electron.
Assume that all of the stored energy is in the form of electrostatic energy:
We have:
Hence
This is one form of the “classical electron radius.”