309-2008-Solutions4

309-2008-Solutions4 - ECE 309 — Electromagnetic Fields...

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Unformatted text preview: ECE 309 — Electromagnetic Fields University of Virginia Fall 2008 Homework # 4 Solutions 1. Energy of a uniformly charged sphere (a) The charge density is given by, ρ = q 4 3 πR 3 = 3 q 4 πR 3 ( * ) so that the energy stored can be calculated as, U = 1 2 Z V ρ Φ dv = ρ 2 Z π sin θdθ Z 2 π dφ Z R r 2 Φ( r ) dr = 2 πρ Z R r 2 Φ( r ) dr From Gauss’s Law, the field due to a uniform sphere of charge is ~ E = q 4 π ˆ r r 2 if r ≥ R ρr 3 ˆ r if r ≤ R Thus, the potential at position r inside the sphere is, Φ( r ) =- Z R ∞ ~ E outside ( r ) · d~ r- Z r R ~ E inside ( r ) · d~ r =- q 4 π Z R ∞ dr r 2- ρ 3 Z r R r dr Φ( r ) = q 4 π 1 R- ρ 6 R 2- r 2 = ρ 3 n 3 R 2 2- r 2 2 o Using relation ( * ) above. Thus the energy stored is, U = 2 πρ 2 3 Z R 3 R 2 2 r 2- r 4 2 dr = 2 πρ 2 3 R 5 2- R 5 10 Thus, U = 4 πρ 2 R 5 15 = 3 q 2 20 π R (b) Using the expression for U in terms of the electric field, U = 2 Z V ~ E · ~ E dv = 4 π 2 n Z R | E inside | 2 dv Z ∞ R | E outside | 2 dv o U = 2 π n ρ 3 2 Z R r 4 dr + q 4 π 2 Z ∞ R dr r 2 o U = 2 π n ρ 3 2 n R 5 5 + R 5 o = 4 πρ 2 R 5 15 , as before (c) Finally, consider a sphere of radius R and total charge Q R . The potential at radius R is given by, Φ R = Q R 4 π R , where Q R = 4 3 ρπR 3 The infinitesimal chage in a spherical layer of thickness dR is, dQ R = 4 πR 2 ρdR and the work required to bring this infinitesimal charge, dQ R from infinity to radius R is, dU = Φ...
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309-2008-Solutions4 - ECE 309 — Electromagnetic Fields...

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