How Does Electric Potential Energy Relate to Capacitors?

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Electric potential energy in capacitors is represented by the formula U = (1/2)CV^2, indicating that only half of the energy supplied during charging is stored, while the other half is lost as heat or radiation. The relationship between charge (Q) and voltage (V) in a capacitor is defined by Q = CV, which clarifies that the simple equation E = QV does not apply directly. The change in energy during charging can be expressed as dU = Vdq = CVdV. The energy imparted to the charge by the battery is QV, but the energy actually stored in the capacitor is only half that amount. Understanding these principles is crucial for accurately analyzing capacitor behavior in electrical circuits.
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all my question is listed in the file below

thanks for your help
 

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1/2 QV is the energy stored in the capacitor. The other half of the energy was dissipated during charging as heat or radiation.
 
Q and V for a capacitor are proportional to each other by Q=CV so the simple E=QV does not apply. The change in energy is dU=Vdq=CVdV.
Then U=\int_0^VCV'dV'=(1/2)CV^2=(1/2)QV.
 
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Clem is, of course, correct. Perhaps I misinterpreted your question (or just answered a different question!) about charging a capacitor with a battery. The energy imparted to the charge by the battery is QV, but the energy stored in the capacitor only half that. Sorry about that!
 
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