How is potential energy is derived from work here?

AI Thread Summary
Potential energy in capacitors is derived from the work done to move charges, represented by the equation PE = 1/2 CV^2 or PE = 1/2 VQ. The confusion arises regarding the sign of potential energy, as it is often thought to be negative due to the relationship between work and potential energy. However, the key lies in the reference point chosen for the system, which affects the sign convention used in calculations. Consistency in the reference point is crucial for correctly deriving the signs in energy equations. Understanding this concept clarifies why potential energy is equated to work without a negative sign.
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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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