ENERGY STORED IN CAPACITOR

ENERGY STORED IN CAPACITOR:-

     We know that capacitor is charge storing device. Its charge storing ability that is capacitance increases in the presence of dielectric medium as explained in the previous topic. Capacitor store its energy inside electric field which can be calculated as follow:                                          

We know that Potential Difference=  Work

                                                     Charge

Or V = Energy

           Charge   

Or Energy = V.Q ----------(i)

By capacitor equation Q=CV-----(ii)

·         When key is open V₁=0 & Q₁=0

·         When key is closed V₂=V & Q₂=Q

Average P.D= V₁+V₂ = 0 + V

                         2           2

We can write V_avg= V  -------(iii)

                              2

Then (i) can be written as

Energy=V_avg x Q----(iv)

Using (iii) & (ii) in (iv) we get

Energy= ( V ) (VC)

                2

Energy=CV²---------(v)

               2
One other way to derive (v) is from graph method which is also shown below

The potential difference across the plates of a capacitor is directly proportional to the charge stored on the plates. This gives a straight line through the origin on a voltage-charge graph. The area under this graph gives the energy stored in a capacitor.

As the area under the graph is a triangle,

area = ½ base x height.

 so Energy = Area=  1 QV

                              2

Whereas Q=CV so

Energy= 1 CV²

                 ²

    Now                   

In case of any dielectric as medium between between plates of capacitor the capacitance is given by

C_med= Aε_o ε_r ----------------- (vi)

              d

also we know tha E= V

                               d

or V= Ed-----(vii)

in case of energy in the presence of dielectric (v) can be written as

Energy = 1  (C_med)(V²)------------(vii)

               2

Putting (vii) & (vi) in (viii) we get

Energy = 1 ( Aε_o ε_r )(Ed)²

                  2           d

after cancellation of d we get

Energy = 1  ε_o ε_r E²(Ad)

              2

Where as Ad= Volume of dielectric so

Energy =  1  ε_o ε_r E²(Volume)

                2

 Energe  = 1  ε_o ε_r E²

Volume     2

As  Energy = Energy density so

     Volume               

Energy density = 1 Aε_o ε_r E²-------(ix)

          2      

 CONCLUSION:-

• Energy density is directly proportional to ε_r
• _{Energy density is directly proportional to} E²