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ECE 3318 Applied Electricity and Magnetism Spring 2017 Prof. David R. Jackson ECE Dept. Notes 26 1 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.) 2 Stored Energy (cont.) Charge formulas: 1 U E v dV 2V 1 U E s dS 2S (volume charge density) (surface charge density) Note: We don’t have a linecharge version of the charge formulas, since a line charge would have an infinite stored energy (as would a point charge). The charge formulas require 0 Electric-field formula: 1 U E D E dV 2V 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. 3 Example Find the stored energy. A m 2 V0 + - r h x We assume an ideal parallel-plate capacitor (no fields outside). V0 ˆ E x h Method #1 V D 0 r xˆ 0 h Use the electric field formula: UE 1 D E dV 2V so 2 1 V U E 0 r 0 dV 2V h 4 Example (cont.) Hence 1 V0 U E 0 r 2 h 2 Ah J We can also write this as 1 A U E 0 r V02 2 h Recall that A C 0 r h Therefore, we have 1 U E CV02 2 5 Example (cont.) Method #2 A m 2 SA Use charge formula: +++++++++++++ 1 U E s dS 2S ------------------ r h x SB (We use the surface-charge form of the formula.) We then have 1 1 U E A s dS B s dS 2 2 SA SB 6 Example (cont.) 1 1 U E AQA B QB 2 2 1 1 A Q A B QA 2 2 1 Q A B 2 so 1 U E QV0 2 A m 2 SA +++++++++++++ r h ------------------ x SB Also, we can write this as 1 U E QV0 2 1 CV0 V0 2 or 1 U E CV02 2 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 2U E C 2 V where UE = stored energy in capacitor V = voltage on capacitor This gives us an alternative method for calculating capacitance. 8 Alternative Capacitance Formula (cont.) Find C (using the alternative formula). A m 2 A r Ex 0 h x B Ideal parallel plate capacitor Assume: E xˆ Ex 0 B h h A 0 0 V VAB E dr Ex dx Ex 0 dx Ex 0 h 9 Alternative Capacitance Formula (cont.) A m 2 A r h Ex 0 x B UE 1 1 1 1 2 2 D E dV E E dV E dV Ex 0 Ah 2V 2V 2V 2 Therefore 1 2 Ex20 Ah 2U E 2 C 2 2 V E h x0 so C A h 10 Example Find the stored energy. a 1 U E D E dV 2V 0 E E r rˆ Er r v 0 C/m 3 Solid sphere of uniform volume charge density (not a capacitor!) Gauss’s Law: r<a 4 3 v0 3 r E rˆ 2 4 0 r r>a 4 3 v0 3 a E rˆ 2 4 0 r 11 Example (cont.) UE 0 2 E dV 2 V 0 2 0 2 0 2 2 E r r dV a V v 0 C/m 3 2 2 2 E r r sin dr d d r 0 0 0 0 2 2 Er2 r r 2 dr 0 Hence we have 2 2 4 3 4 3 a v0 v0 r a 2 2 3 3 U E 2 0 r dr 2 r dr 0 2 2 4 0 r 4 0 r 0 a 12 Example (cont.) Result: v20 4 5 UE a 0 15 Denote This is the energy that it takes to assemble (force together) the charges inside the cloud from infinity. 4 3 Q v 0 a 3 so that 3 v 0 Q 3 4 a We then have 1 Q2 3 UE a 0 20 J 13 Example (cont.) Final result: 1Q UE a 0 2 3 20 a J Q C 0 Note: UE as a0 It takes an infinite amount of energy to make an ideal point charge! 14 Example Find the “radius” of an electron. 1 Q2 3 UE a 0 20 Q2 3 a U E 20 0 Q qe 1.6022 1019 [C] Assume that all of the stored energy is in the form of electrostatic energy: U E mc 2 We have: m 9.1094 1031 [kg] c 2.99792458 10 [m/s] 8 Hence a 1.69 1015 [m] U E mc 2 8.1871 1014 [J] This is one form of the “classical electron radius.” 15