(a) Parallel plate capacitor. A parallel plate capacitor is a combination of two conducting plates held parallel to each other at suitable distance having some dielectric in between them. One of the plate is positively charged and the other is earthed.
Expression for capacitance of parallel plate capacitor
Let a parallel plate capacitor consist of two thin conducting plates A and B held parallel to each other at a certain distance d apart. Medium between plates is given to be vacuum. Plate A is insulated and B is earthed.
When charge +q is given to A, it induces -q on the nearer face of B and +q on the farther face of the plate B. Free +ve charge on B flows to the earth.
Electric field intensity between the plate is
E = \(\frac{\sigma}{ɛ_o}\)
where σ is the charge density and equal to q/A.
Also E = \(\frac{V}{d}\), where V is the P.D. between plates.
V = Ed = \(\frac{\sigma}{ɛ_o}d = \frac{qd}{Aɛ_o}\) [ ∵ \(\sigma = \frac{q}{A}\)]
If C is the capacitance of the parallel plate capacitor, then
C = \(\frac{q}{A} = \frac{q}{\frac {qd}{A\varepsilon_o}} = \frac{A\varepsilon_0}{d}\)
or C = \(\frac{A\varepsilon_0}{d}\)
This is the expression for the capacitance when the medium is air or vacuum
If A is in m2, d is in m, then C is taken in farad. If medium between plates is other than air, capacitance of capacitor is given by
C = \(\frac{\varepsilon_0KA}{d}\)
where K is dielectric constant.
(b) Definition of dielectric constant
In vacuum the capacitance of capacitor,
C2 = \(\frac{\varepsilon_0KA}{d}\) ..........................(ii)
Dividing (i) and (ii), we get
\(\frac{C_2}{C_1} = \frac{\varepsilon_0KA}{d} \times \frac{d}{\varepsilon_0A} = K\)
∴ Dielectric constant of a medium is defined as the ratio of capacitance of capacitor with dielectric as medium to the capacitance of the same capacitor with air as medium.
