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 .