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Questions
Derive an expression for energy stored in a capacitor.
Derive an expression for energy stored in a charged capacitor.
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Solution
Consider a capacitor of capacitance C being charged by a DC source of V volt as shown in figure.
Capacitor charged by a DC source.
During the process of charging, let q' be the charge on the capacitor and V be the potential difference between the plates. Hence
`"C" =("q""'")/"V"`
A small amount of work is done if a small charge dq is further transferred between the plates.
∴ `"dW" ="Vdq"=("q""'")/"Cdq"`
Total work done in transferring the charge
`"W"=int"dw"=int_0^"Q" ("q'")/"C" "dq" = 1/"C"int_0^"Q" "q'" "dq"`
`=1/"C"[(("q""'")^2)/2]_0^"Q" = 1/2 "Q"^2/"C"`
This work done is stored as electrical potential energy U of the capacitor. This work done can be expressed in different forms as follows:
∴ `"U" = 1/2 "Q"^2/"C"=1/2"CV"^2=1/2"QV" (because "Q" = "CV")`
Notes
Students should refer to the answer according to their questions.
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