Concept Review Ch.31 Capacitors

Revision points

Capacitor
A combination of two conductors placed clsoe toeach other is called a capacitor. Oneof the conductors is given a positive charge and the other a negative charge.

Capacitance
For a given capacitor, the charge Q on the capacitor is proportional to the potential difference V between the two plates

So Q α V
or Q = CV

C is called the capacitance of the capacitor.
SI unit of capacitance is coulomb/volt which is written as farad. The symbol F is used for it.
To put equal and opposite charges on the two conductors they may be connected to the terminals of a battery.

Calculation of Capacitance

For parallel plate capacitor

C = ε₀A/d

A = area of the flat plates (each used in the capacitor)
d = distance between the plate

Spherical capacitor

It consists of a solid or hollow spherical conductor surrounded by another concentric hollow spherical conductor.

If inner sphere radius is R1 and Outer sphere radius is R2

Inner sphere is given positive charge and outer sphere negative charge.

C = 4πε₀R₁R₂/[R₂-R₁]

If the capacitor is an isolated sphere (outer sphere is assumed to be at infinity, hence R₂ is infinity and

C = 4πε₀R₁

V becomes Q/C = Q/4πε₀R₁

V = potential

Parallel limit: if both R1 and R2 are made large but R2-R1 = d is kept fixed

we can write
4πR1R2 = 4πR² = A; where R is approximately the radius of each sphere, and A is the surface area of the sphere.

C = ε₀A/d; where A = 4πR1R2 = 4πR²


Cylindrical Capacitor

It consists of a solid or hollow cylindrical conductor surrounded by another concentric hollow cylindrical conductor.

If inner cylinder radius is R1 and Outer cylinder radius is R2 and length is l,
Inner cylinder is given positive charge and outer cylinder negative charge

C = 2πε₀l/ln(R₂/R₁)

Combination of capacitors

Series combination

1/C = 1/C1 + 1/C2 + 1/C3 ...

Parallel combination

C = C1 + C2 + C3

Force between plates of a capacitor

Plates on a parallel capacitor attract each other with a force

F = Q²/2Aε₀

Energy stored in a capacitor

Capacitor of capacitance C has a stored energy

U = Q²/2C = CV²/2 = QV/2

Where Q is the charge given to it.


Dielectric material

In dielectric materials, there are no free electrons. Electrons are bound to the nucleus in atoms. Basically they are insulators. But when a charge is applied, in these materials also atoms or molecules are oriented in a such way that there is an induced. For example, in the case of rectangular slab of a dielectric, if an electric field is applied from left to right, the left surface of the slab gets a negative charge, and the right surface gets positive charge.


Change in capacitance of a capacitor with dielectric in it.

The surface charge density of the induced charge can be related to a measure called Polarization P (which is dipole moment induced per unit volume - where is the dipole? in the diectric slab as the two sides have opposite charges)

If σ_p is the magnitude of the induced charge per unit area on the faces.

The dipole moment (q*Vr(d)) of the slab is then charge*l (distance between faces)
= σ_pAl.
where

A is area of cross section of the dielectric slab


As polarization is defined as dipole moment induced per unit volume,
P = σ_pAl/Al (Al = volume of slab)

= σ_p

The induced surface charge density is equal in magnitude to the polarization P.

Dielectric constant

Because of induced charge, electric field is produced in the slab which is against the field applied on the slab.

Resultant field = Applied field - induced field

Resultant field = Applied field/K

K is greater than 1 and is a constant for give materials. K is called the dielectric constant or relative permittivity of the dielectric.

Dielectric strength

If a very high electric field is created in a dielectric, electrons in valence shell may get detached from their parent atoms and move freely like in a conductor. This phenomenon is called is dielectric breakdown. The electric field at which breakdown occurs is called the dielectric strength of the material.

Capacitance of a parallel plate capacitor with dielectric

C = KC₀
where C₀ is capacitance of a similar capacitor without dielectric.

Because K>1, the capacitance of a capacitor is increased by a factor of K when the space between the parallel plates is filled with a dielectric.

Magnitude of induced charge in term of K

Q_P = Q[1 - (1/K)]

Q_P = induced charge in the dielectric
Q = Applied charge
K = dielectric constant

Gauss's law when dielectric materials are involved

∮KE.dS = Q_free/ε₀ .....(31.14)

Where integration is over the surface, E and dS are vectors, Q_free is the free charge given (charge due to polarisation is not considered) and K is dielectric constant.

The law can also be written as

∮D.ds = Q(free) ...... (31.15)
where D = Eε₀ + P; E and P are vectors
E = electric field and P is polarisation

Electric field due to a point charge placed inside a dielectric

E = q/4πε₀Kr²

Energy in the electric field in a dielectric

u = ½Kε₀E²

Corona discharge

If a conductor has a pointed shape like a needle and a charge given to it, the charge density at the pointed end will be very high. Correspondingly, the electric field near these pointed ends will be very high which may cause dielectric breakdown in air. The charge may jump from the conductor to the air. Often this discharge of charge inot air is accompanied by a visible glow surrounding the pointed end and this phenomenon is called corona discharge.

High voltage generator – Van de Graaff Generator

The apparatus transfers positive charge to a sphere continuously till the potential reaches to around 3*10^ 6 V at which point corona discharge takes place and hence no further charge can be transferred. The charge of course can be increased by enclosing the sphere in a highly evacuated chamber.