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Introductory Physics Homework Help
Energy stored in an electrostatic system
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[QUOTE="Pushoam, post: 5819804, member: 619344"] [COLOR=#ff0000]<Moderator's note: Thread moved from a technical forum, so homework template missing>[/COLOR] To do: To find an expression for energy stored in an electrostatic system with charge density ## \rho ## and volume R. [ I am using R to denote the region filled with the given charge density as I want to keep 'V' reserved for potential ]. By the term "energy of the system " what we mean is : work required to assemble the system So, I have to calculate work required to assemble the system i.e. W. Let's consider the system at the time when its volume is Δ R and I bring charge dq to a point M with position vector ##\vec r ## on its surface. [ATTACH=full]208761[/ATTACH] Work done by me in this process is dW = V(##\vec r ##) dq ##V \left ( \vec r \right ) = k \int _{ΔR}~ \frac { \rho (\vec {r'}) }{r"} d \, τ' ## ## dq = \rho (\vec {r}) dτ ## So, ## dW = k \int _R~ \frac { \rho (\vec {r'}) }{r"} d \, τ' ~ \rho (\vec {r}) dτ ## ## W = \int _R~ \rho (\vec {r}) \{ k \int _{ΔR}~ \frac { \rho (\vec {r'}) }{r"} d \, τ'\} d\,τ ## where ΔR depends on r. [B][COLOR=#b300b3]Is this correct so far?[/COLOR][/B] Another way: W = ½ ##\int_R ~ \rho (\vec r) V (\vec r) d \, τ## ## \rho (\vec r) V (\vec r) = ε_0 (∇⋅ \vec E ) V = ε_0 ( ∇⋅( V \vec E ) - \vec E⋅ ∇ V ) ## ## W = ½ \int_R~ ε_0 ( ∇⋅ (V \vec E ) - \vec E⋅ ∇ V ) d \,τ ## ## = ½ \int_S ~ ε_0 V \vec E ⋅ d \vec a + ½ \int_R ε_0 E^2 d \, τ ## Taking the region of integration to be infinite, (as the charge density outside R is 0), the first integral on R.H.S. becomes 0. Hence, we have, ## W = \frac {\epsilon _0 } 2 \int_{all space} E^2 d \, \tau ##In case of dielectrics, this will be the work required for assembling both free charges and bound charges. For dielectrics, ## \rho = \rho_b + \rho_f,## So, W = ½ ##\int_R ~ \rho (\vec r) V (\vec r) d \, τ## = ½ ##\int_R~ (\rho_b + \rho_f )V (\vec r) d \, τ## W =½ ## \int_R~ \rho_b V d \, τ + ½ \int_R \rho_f V (\vec r) d \, τ## [COLOR=#b300b3][B]Can I take the first term on the R.H.S. as the work required to assemble the bound charges and the second term as the work required to assemble the free charges ?[/B][/COLOR] [B][COLOR=#b300b3] If yes, then,[/COLOR] [/B] ##W_f = ½ \int_R \rho_f V (\vec r) d \, τ \\ \rho_f=∇.\vec D## And doing some further calculation, ## W_f = ½ \int_{all space} \vec D . \vec E ~d \, τ ## [COLOR=#b300b3][B]Is this correct so far?[/B][/COLOR] [/QUOTE]
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Energy stored in an electrostatic system
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