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Capacitors: Types, Expressions, Effect of Insertion

What is a capacitor? Describe different types of capacitors and derive expressions for their capacitances. Derive expression for the energy stored in a parallel plate capacitor. Discuss the effect of insertion of a dielectric medium between the plates of a capacitor.

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Understanding Capacitors

1. What is a capacitor?:

Capacitor is a device which stores electrostatic energy and is widely used in electric and electronic circuits. In general, a capacitor is an arrangement of two conducting bodies, called plates separated by a small distance between them. In principle these bodies can be of any shape (though in practice simple shapes such as plates, spheres or cylinders are used). The gap between the two bodies can have air or any other dielectric medium. If a charge +Q is given to one of the plates and -Q to the other (say by connecting the two plates to a battery momentarily), the charges will spread themselves on the surfaces of the respective plates. The plates will be equipotential surfaces, each having a voltage (say V1 and V2). There will thus be a potential difference V = V1 - V2 between the plates and an electrostatic field will be set up between them.

2. Capacitance of a capacitor:

When we put a certain charge +Q on one of the plates and -Q on the other, the potential difference that develops between the plates shall have a unique value for a given capacitor and the same is determined by the construction and dimensions of the capacitor. Thus we can say that the ratio of the charge on each plate and the voltage developed between the plates is a constant for a given capacitor. This constant has been given the name "Capacitance of the Capacitor" and it is represented by symbol C. Thus

Capacitance C = Q/V Coulombs/Volt

The units of capacitance are Coulombs per Volt and the same is also called Farad. Thus 1 Coulomb/Volt = 1 Farad

For a given potential difference between the plates, higher the charge Q that can be stored on the capacitor plates, higher the capacitance C. Thus, capacitance is a measure of the charge storing capacity of a capacitor.

3. Capacitance of a parallel plate capacitor:

Let us consider a capacitor comprising of two parallel metallic plates each with area A and separated by a distance d. Let the charge density on each plate be σ c/m2 (on one plate +ve and on the other -ve) and potential difference between the plates V.

+σ c/m2 Area A
Pot. Diff. V
Gap d
-σ c/m2

The electric field intensity between two infinite plates each of which carries a charge density of σ (one plate +ve and the other -ve) is given by :

E = σ ...(1)
ε0

Though the above expression is for infinite sheets, we can use the same approximately if the dimensions of the sheets are >> d (i.e.A >> d2).

Since electric field intensity between the plates is also defined as the potential gradient, E = V/d ...(2)

And total charge Q on each plate = σ . A => σ = Q / A ...(3)

Substituting above values for E and σ in (1) we get :

V = Q => Q = ε0 A
d Aε0 V d

By definition Q/V is the capacitance; hence, capacitance of a parallel plate capacitor is given by: C = ε0 A/d ...(4)

4. Capacitance of a spherical capacitor:

Let us consider a spherical capacitor comprising of two concentric, hollow spheres with radius ra and rb. Let the outer sphere be given +Q charge and the inner sphere -Q charge which spread uniformly on the respective sphere surfaces.

+Q

rb
-Q
ra

We know the potential of a sphere (on the surface and inside) of radius R carrying a charge Q is given by: V = Q/4Πε0R and outside at a distance r from the centre: V = Q/4Πε0 r

Thus, potential of inner sphere for r < ra (surface and inside) Va = - Q/4Πε0ra and for r > ra , Va= - Q/4Πε0 r .

Potential of outer sphere for r < rb (surface and inside) Vb = Q/4Πε0rb and for r > rb , Vb= Q/4Πε0 r .

Thus, net potential on the surface and inside the inner sphere (r < ra) = Q/4Πε0rb - Q/4Πε0ra = Q/4Πε0 (1/rb - 1/ra).

Net potential at a distance r from the centre, outside the outer sphere (r > rb) = Q/4Πε0r - Q/4Πε0r = 0

Net potential in the space between the two sphere (ra < r < rb) = Q/4Πε0rb - Q/4Πε0r = Q/4Πε0 (1/rb - 1/r).

Thus, in the space between the two spheres, the potential varies between Q/4Πε0 (1/rb - 1/ra) (for r = ra at the inner sphere) and zero (for r = rb at the outer sphere).

The potential difference between the two spheres = V = Q/4Πε0 (1/rb - 1/ra)

Capacitance of a spherical capacitor C = Q/V = 4ε0 (ra.rb/(rb - ra) ...(5)

5. Capacitance of cylindrical capacitor:

Let us first derive a general expression for field in a hollow charged cylinder with radius R and length l and charge density σ.

R

Charge density +σ
Gaussian
l surface r>R

Let us first consider as Gaussian surface a cylinder with radius r such that r > R. As per Gauss theorem we can write :

E (r) S = Q/ε0 => E (r) 2r.l = Q/ε0

E (r) = Q/2ε0 l r (direction of field vector being radially outwards)

Next we consider the case of r < R. Since the charge enclosed within the Gaussian surface is zero, the field with in the cylinder shall be zero.

Let us consider a capacitor with two concentric hollow ...

Solution Summary

This solution provides detailed explanations with diagrams and equations for various questions regarding capacitors.

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