Physics Capacitors

CAPACITORS

A combination of two conductors separated by an insulator is called a capacitor (Fig1). The conductors have charges, say `Q_1` and `Q_2`, and potentials `V_1` and `V_2`. Usually, in practice, the two conductors have charges Q and & Q, with potential difference `V = V_1 & V_2` between them. We shall consider only this kind of charge configuration of the capacitor. (Even a single conductor can be used as a capacitor by assuming the other at infinity.) The conductors may be so charged by connecting them to the two terminals of a battery. Q is called the charge of the capacitor, though this, in fact, is the charge on one of the conductors the total charge of the capacitor is zero.

The simplest example of a capacitor consists of two conducting plates of area A, which are parallel to each other, and separated by a distanced, as shown in Figure2.
Experiments show that the amount of charge Q stored in a capacitor is linearly proportional to the electric potential difference between th e plates. Thus, we may write

`Q = C |DeltaV|`

where C is a pos itive proportionality constant called capacitance. Physically, capacitance is a measure of the capacity of storing electric charge for a given potential difference~ V. The Sl unit of capacitance is the farad(F):

1 F = 1 farad = 1 coulomb / volt = 1 C/V

A typical capacitance is in the picofarad `( 1 pF = 10^-12 F)` to millifarad range,` ( 1 mF = 10^-3 F = 1000 `mu` F ; 1 mu F = 10^-6 F).`

Figure 3(a) shows the symbol which is used to represent capacitors in circuits. For a polarized fixed capacitor which has a definite polarity, Figure 3(b) is sometimes used.

Parallel Plate Capacitor

A parallel-plate capacitor consist of two large plane metallic plates A and B each of area A separation `d`. Plate A is positively charged and plates B is earthed. If K is the dielectric constant of the material medium and E is the field that exist between the two plates, then

`E = sigma/epsilon = sigma/(kepsilon_0)`

`V/d = q/(Kepsilon_0A),` `{ E = V/d `and` sigma= q/A }`

`C=q/V = (Kepsilon_0A)/d`

If medium between the plates is air or vaccume , then K=1

`C_0 = epsilon_0A/d`

`text(Case 1 :)`

`text(When space between the parallel plates )`
`text(is partially filled with a dielectric of thickness t),`

If no slab is introduced between the plates of the capacitors, then a field `E_0` given by `E_0 = sigma/epsilon_0,` exists in a space d.

on inserting t and a field `E = E_0/K` exists inside the slab of thickness t, a field `E_0` exists in remaining space (d-t). If V is the total potential

`V = E_0(d-t) + Et`

`V = E_0[d-t+(E/E_0)t]`

But `E_0/E = K` = Dielectric constant

`V = sigma/epsilon_0[d - t + t/k]`

`V = q/(epsilon_0A)[d - t + t/k]`

`C = q/V = epsilon_0A/(d-t(1-1/K))`

So, on including a dielectric slab of thickness t and dielectric constant K the capacitance increases by the same amount as the effective air spacing between the plates is made d - t(1 - `1/K`)

`text(Case 2 :)`

`text(When the space between the parallel plate)`
`text( capacitor is partly filled by a conducting slab of thickness t),`

If no conducting slab is introducing between the plates, then a field `E_0 = sigma/epsilon_0` exists in a space d. If `C_0` be the capacitance then,

`C_0 = (epsilon_0A)/d`

on inserting the slab, field inside it is zero and so a field `E_0 = sigma/epsilon_0` now exists in a space (d-t)

`V = E_0(d-t)`

`V = sigma/epsilon_0(d-t)`

`V = q/(Aepsilon_0)(d-t)`

`C = q/V = epsilon_0A/(d-t)`

`C = (epsilon_0)A/d(1 - t/d)`

`C = C_0/(1 - t/d)`

Since `d-t < d` , `C > C_0`

Capacitance increases on insertion of conducting slab between the plates of capacitor.

If t=d, then `C->oo`, if a conducting slab occupies the complete space between the plates of the capacitor, then `C->oo`

Spherical Capacitor

A spherical capacitor consists of a solid or hollow spherical conductor surrounded by another concentric hollow spherical conductor.
let's consider a spherical capacitor which consists of two concentric spherical shells of radii `a` and `b`, as shown in Figure. The inner shell has a charge `+Q` uniformly distributed over its surface, and the outer shell an equal but opposite charge `-Q`

The potential difference between the two conducting shells is :

`DeltaV = V_b - V_a = -int_a^b E_r dr = - Q/(4pi epsilon_0)int_a^b dr/r^2 = - Q/(4pi epsilon_0)(1/a - 1/b) `

`= - Q/(4pi epsilon_0)(b-a)/(ab)`

which yields

`C = Q/|DeltaV| = 4pi epsilon_0 (ab)/(b-a)`

Again, the capacitance C depends only on the physical dimensions, a and b. An " isolated" conductor(with the second conductor placed at infinity) also has a capacitance. In the limit where `b -> oo,` the above equation becomes.

`lim_(b->oo) C = lim_(b->oo) 4pi epsilon_0 (ab)/(a-b) = lim_(b->oo) 4pi epsilon_0 a/(1-a/b) = 4 pi epsilon_0a`

Thus, for a single isolated spherical conductor of radius R, the capacitance is

`C = 4pi epsilon_0R`

The above express ion can also be obtained by noting that a conducting sphere of radius R with a charge
Q unifonnly distributed over its surface has `V = Q/4pi epsilon_0 R,` using infinity as the reference point having
zero potential, `V(oo) ~ 0.` This gives

`C = Q /|DeltaV| = Q/(Q/(4pi epsilon_0R)) = 4pi epsilon_0R`

As expected, the capacitance of an isolated charged sphere only depends on its geometry, namely, the
radius R.

Cylindrical Capacitor

It consist of two coaxial metallic cylindrical of radius a and outer radius b. The outer surface of the cylindrical of radius b is earthed. The space between the two cylinders is filled with material of dielectric constant K. Let inner cylinder be given a charge per unit length of `lamba (=q/l)`. A charge `-q` is induced on length l at inner surface of outer cylinder

`E = lambda / (2 epsilon_0 r)` for a < r < b

`(dV) / (dr) = lambda /(2pi epsilon_0)Kr`

`int_text(inner surface)^(outer surface) dV = - lambda/(2pi epsilon_0 K) int_(r=a)^(r=b) (dr)/r`

`V_b - V_a = -lambda/(2pi epsilon_0 K) ln(b/a)`

Since, inner surface is at higher potential and outer at lower potential so

`Delta V = q/(2pi epsilon_0LK) ln(b/a)`

`C = q/|DeltaV| = (2pi epsilon_0 LK)/ (ln (b/a))`