How is potential energy is derived from work here?

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SUMMARY

The discussion focuses on the derivation of potential energy from work in capacitors, specifically addressing the relationship between work done and potential energy stored. The formula for potential energy (PE) in a capacitor is established as PE = 1/2 CV^2, derived from the work done to move charges, represented by the integral of voltage over charge. A key point raised is the confusion regarding the sign of potential energy, with the clarification that the reference point chosen affects the sign convention used in calculations. Consistency in reference points is essential for accurate derivation.

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kiwibird4
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Since capacitors can store electrical energy in the form of a electrostatic field between its plates, one can derive that potential energy by looking at the work it takes to "push" the charges around to create the voltage.

Therefore Work =
integral from 0 to Q of V(q) dq =
integral from 0 to Q of (1/C)q dq =
Q^2/2C =
1/2 CV^2=
1/2 VQ which consequently equals the potential energy stored in the capacitor

so my question overall is::
Why it is not negative 1/2 VQ for PE since PE is the negative of work?
Everywhere I look there is never a -PE=W but they immediately equate the two being PE=W. I think I am missing something very simple here.
 
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The whole thing in this case, is the reference point you take. This can either be you (or whatever produces work for that matter) and system ( the capacitor in this case). You must be consistent with what you choose and use in order to derive the appropriate signs.
 

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