1. 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.
2. For parallel plate capacitor
C = ε0A/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 = 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
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 = 2πε0l/ln(R2/R1)
5. Combination of capacitors
1/C = 1/C1 + 1/C2 + 1/C3 ...
C = C1 + C2 + C3
6. Force between plates of a capacitor
Plates on a parallel capacitor attract each other with a force
F = Q²/2Aε0
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 σp 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)
A is area of cross section of the dielectric slab
As polarization is defined as dipole moment induced per unit volume,
P = σpAl/Al (Al = volume of slab)
The induced surface charge density is equal in magnitude to the polarization P.
9. 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
QP = Q[1 - (1/K)]
QP = induced charge in the dielectric
Q = Applied charge
K = dielectric constant
10. Gauss's law when dielectric materials are involved
∮KE.dS = Qfree/ε0
Where integration is over the surface, E and dS are vectors, Qfree 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 = Eε0 + 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/4πε0Kr²
Energy in the electric field in a dielectric
u = ½Kε0E²