Stored energy in the electric field continued where

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Stored Energy in the Electric Field (continued) where the vector identity, , has been used. Next, the divergence theorem is used on the first term, replacing the volume integral by an integral over the surface that surrounds the volume:
Stored Energy in the Electric Field (continued) We now have: in which the region of integration now includes all space, or wherever the field and potential exist. We are no longer constrained to the volume taken up by the charge. This means that the surface of integration in general lies at infinity, or at an infinite radius from the otherwise compact charge.
Stored Energy in the Electric Field (continued) At the infinite distance, the potential and Dfields begin to resemble those of a point charge: This means that the surface integral will vanish, because the inverse cube dependence in the integrand falls off at a more rapid rate with r than the surface area increases (surface area increases only as the square of the radius). and therefore:
Stored Energy in the Electric Field (continued) This means that the surface integral will vanish, because the inverse cube dependence in the integrand falls off at a more rapid rate with r than the surface area increases (surface area increases only as the square of the radius).
Electric Field Energy and Energy Density The field energy expression now reads: but we know that: which leads us to the final result: where the energy density in the electric field is defined as:
Example Find the energy stored in free space for the region, 0 < ρ< a, 0 < Φ< π, 0 < z < 2, given the potential field V =: (a) Vo ρ/a; (b) Vo (ρ/a)cos2Φ.
Solution (a) 220200/1aVEaVEaVEazzVaVaVVE
Solution (a) 200200022200002022000571.122221VWzaVWdzddaVWEdvWEaEaEvolE  
Solution (b) 422032202222022000202020cossincos4cossin4coscossin2)sin(cos)2(coscos)/(1cos)/(1aVaVaVEaVaVEaVaVEaVaVEazzVaVaVVE
Solution (b)           aaaEaEaEvolEdzddaVdzddaVdzddaVWdzddaVaVaVaVWdzdddvbutdvaVaVaVaVWdvEW0020422000020322000020222000020422032202220220022200204220322022220220020cos2sincos242sin2cossincos42sin2;2sincossin4;cossincos4cossin4221
Solution (b) 04cos224sincos24:7854.04sin41224cos1212sin;2sin2:200402220000203220020020000222002002022200    zaVdzddaVForVzaVdzddaVForaaaa
Solution (b) 20020020020020000002220024002042200374.15890.007854.0

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