emsol03 - Electromagnetic Theory I Solution Set 3 Due 13...

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Unformatted text preview: Electromagnetic Theory I Solution Set 3 Due: 13 September 2010 9. The formula for the total energy stored in a static electric field U = ( ǫ / 2) integraltext d 3 x E 2 includes the self-energy of the charges as well as their mutual potential energy. As discussed in class the self-energy of a point charge is infinite. Study this issue for a single charge Q by spreading out a point charge in two different ways: a) Replacing the point charge by a uniformly charged spherical surface of radius δ and b) Replacing the point charge by a uniformly charged ball of radius δ . In each case, find the electric field everywhere and evaluate the integral defining U . Classical physics puts no apriori limit on how small δ could be. However, relativistic quantum mechanics implies that a particle’s location cannot be known within a distance smaller than its Compton wavelength planckover2pi1 /mc . For cases a) and b), assume δ = planckover2pi1 /mc to get an estimate for the self energy of the electron. Solution : For r > δ , the electric field is E > = Q ˆ r/ 4 πǫ r 2 in both cases a) and b). So for this region we have U > = ǫ 2 integraldisplay r ≥ δ d 3 x Q 2 16 π 2 ǫ 2 r 4 = 4 π integraldisplay ∞ δ dr Q 2 32 π 2 ǫ r 2 = Q 2 8 πǫ integraldisplay ∞ δ dr r 2 = Q 2 8 πǫ δ (1) a) In this case, E = 0 for r < δ , so the self energy for this model is U a = U > = Q 2 8 πǫ δ (2) For the electron Q =- e , so setting δ = planckover2pi1 /m e c we get the estimate E self ≈ e 2 m e c/ 8 πǫ planckover2pi1 = m e c 2 α/ 2. b) In this case, the electric field for r < δ is E < = Q r / (4 πǫ δ 3 ), and the energy for this region is U < = ǫ 2 integraldisplay r ≤ δ d 3 x Q 2 r 2 16 π 2 ǫ 2 δ 6 = 4 π integraldisplay ∞ δ dr Q 2 32 π 2 ǫ δ 6 = Q 2 8 πǫ integraldisplay δ r 4 dr δ 6 = Q 2 40 πǫ...
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This note was uploaded on 09/25/2011 for the course PHY 6346 taught by Professor Staff during the Spring '08 term at University of Florida.

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emsol03 - Electromagnetic Theory I Solution Set 3 Due 13...

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