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# Energy stored in a capacitor and energy density .

In this topic we will discuss about the Energy stored in a capacitor and energy density, when a capacitor is charged from zero to Q .

ENERGY STORED IN A CAPACITOR –

Suppose the plates of the capacitor is initially uncharged , and the charge is to be transferred from plate 2 to plate 1 , at a small time charge dQ transferred and at the final time it reaches from 0 to Q .

Let V is the potential difference across the plates then work done for small instant is

dW = V . dQ  = (Q/C)dQ .

integrating both sides,

∫ dW =∫(Q/C)dQ from limit 0 to Q ,  then we get ,

net work done W = Q2/2C ,

also as we know Q=CV,  so work done may be written as ,

W = potential energy stored = U =( ½) CV2   = ½ QV .

If we plot the graph  potential v/s charge we get,

***Similarly we can plot the graph (i)  U v/s V ( when C is constant )

(ii) U v/s C (when V is constant)

(iii) U v/s V  (when Q is constant)

Total energy stored in the combination of capacitors

In  series combination –  U = Q2/2CS But in series combination 1/Cs = 1/C1  + 1/C2 + 1/C3   ….

So we can write in series combination total energy U = U1 + U2 + U3 + …..

In parallel combination – U = ½ Cp V2 = ½ C1 V2 + ½ C2 V2 + ½ C3V2 …..

So we can write in parallel combination total energy of the combination   U = U1+ U2 + U3 ……

Energy density of a  parallel plate capacitor

Energy density of the capacitor is the energy stored per unit volume of the capacitors or condensers . We can write energy density u = U/ ( volume )

u = ( ½ C V2 )/ A d  ( Where A is the area of the plate and D is the separation between the plates .

But potential V = E( electric field) d  . putting this value in the above equation we get  ,

u = ½ ( ϵ0 A /d ) ( E2 d2 / Ad ) = ½ ϵ0 E2 .

the  unit of energy density is  J/m3 .

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