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Energy stored in capacitor





7. Energy stored in a capacitor



  • Consider a capacitor of capacitance C, completely uncharged in the begning.
  • Charhing process of capacitor requires expanditure of energy because while charging a capacitor charge is transferred from plate at lower potential to plate at higher potential.
  • Now if we start charging capacitor by transporting a charge dQ from negative plate ti the positive plate then work is done against the potential difference across the plate.
  • If q is the amount of charge on the capacitor at any stage of charging process and φ is the potential difference across the plates of capacitor then magnitude of potential difference is φ=q/C.
  • Now work dW required to transfer dq is
    dW=φdq=qdq/C
  • To charge the capacitor starting from the uncharged state to some final charge Q work required is
    Integrating from 0 to Q
         W=(1/C)∫qdq
         =(Q2)/2C      
    (14a)
         =(CV2)/2
         =QV/2


    Which is the energy stored in the capacitor and can also be written as
         U=(CV2)/2 ---(15)

  • From equation 14c,we see that the total work done is equal to the average potential V/2 during the charging process ,multiplied by the total charge transferred
  • If C is measured in Farads ,Q in coulumbs and V in volts the energy stored would in Joules
  • A parallel plate capacitor of area A and seperation d has capacitance

    C=ε0A/d

  • electric field in the space between the plates is
    E=V/d or V=Ed

    Putting above values of V and C in equation 14b we find
    W=U=(1/2)(ε0A/d)(Ed)2
    =(1/2)ε0E2(Ad)
    =(1/2)ε0E2.V
    ---(16)

  • If u denotes the energy per unit volume or energy density then
    u=(1/2)ε0E2 x volume
  • The result for above equation is generally valid even for electrostatic field that is not constant in space.





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