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.

**2. For parallel plate capacitor**

C = ε

_{0}A/d

A = area of the flat plates (each used in the capacitor)

d = distance between the plate

**3. 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 = 4πε

_{0}R

_{1}R

_{2}/[R

_{2}-R

_{1}]

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

_{2}is infinity and

C = 4πε

_{0}R

_{1}

V becomes Q/C = Q/4πε

_{0}R

_{1}

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 = ε

_{0}A/d; where A = 4πR1R2 = 4πR²

**4. Cylindrical Capacitor**

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 = 2πε

_{0}l/ln(R

_{2}/R

_{1})

**5. Combination of capacitors**

Series combination

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

Parallel combination

C = C1 + C2 + C3

6.

**Force between plates of a capacitor**

Plates on a parallel capacitor attract each other with a force

F = Q²/2Aε

_{0}

7.

**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.

8. Change in capacitance of a capacitor with dielectric in it.

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

_{p}Al.

where

A is area of cross section of the dielectric slab

As polarization is defined as dipole moment induced per unit volume,

P = σ

_{p}Al/Al (Al = volume of slab)

= σ

_{p}

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

9.

**Capacitance of a parallel plate capacitor with dielectric**

C = KC

_{0}

where C

_{0}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

10.

**Gauss's law when dielectric materials are involved**

∮K

**E**.

**dS**= Q

_{free}/ε

_{0}

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)

where D = Eε

_{0}+ P; E and P are vectors

E = electric field and P is polarisation

11. Electric field due to a point charge placed inside a dielectric

E = q/4πε

_{0}Kr²

Energy in the electric field in a dielectric

u = ½Kε

_{0}E²

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