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# notes24 2317 - ECE 2317 Applied Electricity and Magnetism...

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Prof. Filippo Capolino ECE Dept. Fall 2006 Notes 24 ECE 2317 ECE 2317 Applied Electricity and Magnetism Applied Electricity and Magnetism Notes prepared by the EM group, University of Houston . (used by Dr. Jackson, spring 2006) Stored Energy Stored Energy 1 2 Ev V Ud V ρ 1 2 E V UD E d V =⋅ () 0 Φ∞= Note: Please see the text book or supplementary notes for a derivation. 1 2 Es S S charge formula: electric-field formula: (volume charge density) (surface charge density) Example + - ε r h x V 0 A [m 2 ] Method #1 ± ± 0 0 0 2 0 0 1 2 1 2 E V r Er V E d V V Ex h V Dx h V V h εε ⎛⎞ = ⎜⎟ ⎝⎠ = = Find the stored energy Example (cont.) Example (cont.) [] 2 0 0 1 J 2 V UA h h = 2 00 1 2 A UV h = Or, 2 0 1 2 E UC V = Recall that 0 r A C h = Hence

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Example (cont.) Example (cont.) Method #2 1 2 1 2 11 22 AB Ev V s S As Bs SS Ud V dS dS dS ρ ρρ ∫∫ (since we have surface charge in this problem) ε r h x A [m 2 ] A B + + + + + + + + + + + + + ------------------ Example (cont.) Example (cont.) () 1 2 E AA BB A B UQ Q QQ Q =Φ +Φ −Φ Φ 0 00 1 2 1 2 E V CV V = = 0 1 2 E V = 2 0 1 2 E UC V = Example Example v 0 [C/m 3 ] 0 a Method #1 : 1 2 E V UD E d V =⋅ Gauss’s Law: r < a 3 0 2 0 4 3 4 v a Er r ρπ πε ⎛⎞ ⎜⎟ = ⎝⎠ ± 3 0 2 0 4 3 4 v r r = ± r > a Find the stored energy Solid sphere of uniform volume charge density Example (cont.) Example (cont.) ( ) ( ) 2 2 2 0 000 0 0 33 0 sin 2 2 44 VV r r a vv E a UE d V E d V r d r dd r d r ra Ur d r r d r rr ππ θθ φ π == = = =+ ∫∫∫ v 0 [C/m 3 ] 0 a
Example (cont.) Example (cont.) Result: 2 5 0 0 4 15 v E Ua ρ π ε ⎛⎞ = ⎜⎟ ⎝⎠ Let: 3 0 0 3 4 3 3 4 v v Qa Q a ρπ = = as 0 E U a →∞ [] 2 0 13 J 20 E Q U a επ = Note: KCL Law KCL Law

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notes24 2317 - ECE 2317 Applied Electricity and Magnetism...

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