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

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.

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

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

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

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

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.

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

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

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:

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.”