IIT JEE Physics Formula Revision - Ch. 31. Capacitors

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

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 = Q²/2Aε₀

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 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.

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

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

∮KE.dS = Q_free/ε₀

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 = 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