1. Applied Electricity and Magnetism
ECE 2317
Applied Electricity and Magnetism
Fall 2014
Prof. David R. Jackson
ECE Dept.
Notes 26

2. Stored Energy The charge-density formulas require
Charge formula:
(volume charge density)
(surface charge density)
The charge-density formulas require
Electric-field formula:
Note:
We don’t have a line-charge version of the charge formulas, since a line charge would have an infinite stored energy.
Please see the textbook for a derivation of these formulas.

3. Example A [m2] x r h Method #1 + V0 Find the stored energy- r h x V0
A [m2]
Find the stored energy
Method #1
Use electric field formula: so

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

5. Example (cont.) Method #2 A [m2] r h x A Use charge formula:B
Use charge formula:
(We use the surface charge form of the formula.)
We then have

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

7. 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.

8. Solid sphere of uniform volume charge density (not a capacitor!)
Example
Find the stored energy
v0 [C/m3] 0 a
Solid sphere of uniform volume charge density (not a capacitor!)
Gauss’s Law:
r < a
r > a

9. Example (cont.)
v0 [C/m3] 0 a
Hence we have

10. Example (cont.) Result: Denote so that We then have Note:
This is the energy that it took to assemble (force together) the charges from infinity.
Denote
so that
We then have
Note:

11. Example Find the “radius” of an electron. Also, Hence
Assume that all of the stored energy is in the form of electrostatic energy.
Also,
Hence
This is one form of the “classical electron radius.”