Derive an expression for the capacitance of parallel plate capacitor.
Answers
If two conductors , which are insulated from each other, are holding charges +Q and -Q respectively on their surfaces
and if the potential difference between them is V, Then the capacitance C of the system of these conductors is defined as C = Q/V
Parallel plate capacitor with dielectric
Let us assume, we have two large plates, each of area A, separated by a distance d. The charge on the plates is ±Q,
corresponding to the charge density ±σ (with σ = Q/A).
When there is vacuum between the plates, field E0 inside capacitor plates is given by, begin mathsize 12px style E subscript o space equals space sigma over epsilon subscript o end style ......................(1)
Consider next a dielectric inserted between the plates fully occupying the intervening region. The dielectric is polarised by the field E0,
hece the effect is equivalent to two charged sheets (at the surfaces of the dielectric normal to the field) with surface charge
densities σp and –σp. The electric field in the dielectric then corresponds to the case when the net surface charge density on the
plates is ±(σ – σp).
That is, begin mathsize 12px style E space equals space fraction numerator sigma space minus space sigma subscript p over denominator epsilon subscript o end fraction end style .............................(2)
so that the potential difference across the plates is begin mathsize 12px style V space equals space E d space equals space fraction numerator sigma space minus space sigma subscript p over denominator epsilon subscript o end fraction d end style ...........................(3)
For linear dielectrics, we expect σp to be proportional to E0, i.e., to σ. Thus, (σ – σp) is proportional to σ and we can write
Thus, (σ – σp) is proportional to σ and we can write begin mathsize 12px style sigma space minus space sigma subscript P space equals space sigma over K end style ................................(4)
where K is a constant characteristic of the dielectric. Since K > 1 We then have begin mathsize 12px style V space equals space fraction numerator sigma space d over denominator epsilon subscript o K end fraction space equals space fraction numerator Q space d over denominator A space epsilon subscript o K end fraction end style ......................(5)
The capacitance C, with dielectric between the plates, is then begin mathsize 12px style C space equals space Q over V equals fraction numerator epsilon subscript 0 space K space A over denominator d end fraction end style ........................(6)
The product ε0K is called the permittivity of the medium and is denoted by ε