- A capacitor is a device that stores electric charge and consists of two conductors carrying equal but opposite charges. There is a potential difference between the conductors. - The amount of charge Q stored in a capacitor is proportional to the potential difference ΔV between the conductors. The constant of proportionality is called the capacitance C. - Inserting a dielectric material between the capacitor plates increases the capacitance by a factor called the dielectric constant. Within the dielectric, the electric field is reduced even though the surface charges remain the same.

1. Capacitors and dielectrics
Dimpal Boro
Tezpur University
November 4, 2022

2. Capacitors
A capacitor is a device which stores electric charge. Capacitors
vary in shape and size, but the basic configuration is two
conductors carrying equal but opposite charges.
Figure: Basic configuration of a capacitor

3. Potential difference between the two conductors
There is a potential difference between the two conductors of the
capacitor, with the positively charged conductor at a higher
potential than the negatively charged conductor.
If V+ is the electric potential on the positively charged conductor
and V− is the electric potential on the negatively charged
conductor, the electric potential difference between the two
conductors is
∆V = V+ − V− =
Z (−)
(+)
⃗
E.dl

4. Capacitance of a capacitor
Experiments show that the amount of charge Q stored in a
capacitor is linearly proportional to ∆V , the electrical potential
difference between the conductors.
Thus, we may write
Q = C∆V
Where C is a positive proportionality constant called capacitance.
Physically, capacitance is a measure of the capacity of storing
electric charge for a given potential difference ∆V .
The SI unit of capacitance is the farad(F).
1F = 1coulomb/volt = 1C/V

5. Parallel-plate capacitors
The simplest example of a capacitor consists of two conducting
plates of area A, which are parallel to each other, and seperated by
a distance d. Such capacitors are called parallel-plate capacitors.
Figure: A parallel-plate capacitor
Figure: The electric field between the plates of a parallel-plate capacitor

6. Capacitance of a parallel-plate capacitor
The potential difference V between the plates of the capacitor
V = Ed =
σ
ϵ0
d =
d
ϵ0A
Q
C =
Q
V
C =
ϵ0A
d
Figure: A parallel plate capacitor

7. Capacitance of a real parallel-plate capacitor
The formula
C =
ϵ0A
d
is not exact, because the field is not really uniform everywhere
between the plates, as we assumed. The charge density rises
somewhat near the edges of the plates and thus the capacity of the
plates is a little higher than we computed.
Figure: Electric field at the edge of a parallel-plate capacitor

8. Work done to charge a capacitor
To ”charge up” a capacitor, we need to remove electrons from the
positive plate and carry them to the negative plate.
dW = (
q
C
)dq
The total work necessary, then, to go from q = 0 to q = Q, is
W =
Z Q
0
q
C
dq =
1
2
Q2
C
W =
1
2
CV 2
Figure: Work is done by an external agent in bringing +dq from the
negative plate and depositing the charge on the positive plate.

9. High-voltage breakdown
Figure: The electric field near a sharp point on a conductor is very high.

10. High-voltage breakdown
Figure: The field of a pointed object can be approximated by that of two
spheres at the same potential.
Potential of the larger ball
ϕ1 =
1
4πϵ0
Q
a
Potential of the smaller ball
ϕ2 =
1
4πϵ0
q
b
But ϕ1 = ϕ2, so
Q
a
=
q
b

11. High-voltage breakdown
Figure: The field of a pointed object can be approximated by that of two
spheres at the same potential.
Ea
Eb
=
Q/a2
q/b2
=
b
a
Therefore the field is higher at the surface of the small sphere. The
fields are in the inverse proportion of the radii.

12. Dielectrics
Michael Faraday showed experimentally that the capacitance of a
capacitor is increased when an insulator is put between the plates
of a capacitor.
Figure: Dielectric material between capacitor plates
The capacitance is increased by a factor κ and it is called the
dielectric constant.

13. Inside dielectrics
Capacitance of a capacitor is
C =
Q
V
When we put a piece of insulating material between the plates, we
find that capacitance is larger. That means, of course, that the
voltage is lower for the same charge.
Voltage difference is the integral of the electric field across the
capacitor; so we must conclude that inside the capacitor, the
electric field is reduced even though the charges on the plates
remain unchanged.

14. Charged induced on the surface of a dielectric
Gauss’s law tells us that the flux of the electric field is through a
closed surface is directly related to the enclosed charge.
Figure: A parallel-plate capacitor with a dielectric. The lines of E are
shown.

15. A model of a dielectric
Figure: If we put a conducting plate in the gap of a parallel-plate
condenser, the induced charges reduce the field in the conductor to zero.
Figure: A model of a dielectric: small conducting spheres embedded in an
idealized insulator.

16. Polarization
The idea of regions of perfect conductivity and insulation is not
essential.
Each of the small spheres acts like a dipole, the moment of which
is induced by the external field.
The only thing that is essential to the understanding of dielectrics
is that there are many little dipoles induced in the material.

17. Possible mechanism of polarization
Figure: An atom in an electric field has its distribution of electrons
displaced with respect to the nucleus.

18. Polarization vector P
In each atom there are charges q seperated by a distance δ, so that
qδ is the dipole moment per atom.
If there are N atoms per unit volume, there will be a dipole
moment per unit volume equal to Nqδ.
This dipole moment per unit volume is represented by a vector, P.
It is in the direction of individual dipole moments, i.e., in the
direction of charge seperation δ:
P = Nqδ

19. Calculation of the charged induced on the surface of a
dielectric
Figure: a long string of dipoles
Figure: (a) A cylinder with uniform dipole distribution. (b) Equivalent
charge distribution.

20. Calculation of the charged induced on the surface of a
dielectric
Figure: An uniformly polarized sphere in an electric field
Figure: A sphere of lined-up molecular dipoles (a) is equivalent to
superposed, slightly displaced, spheres of positive (b) and negative (c)
charges.

21. Calculation of the charged induced on the surface of a
dielectric
The electric dipole moment QP produces is QPh and it is equal to
the total electric dipole moment of all the little electric dipoles.
Total dipole moment = P(Ah)
QPh = PAh
QP = PA
Surface charge density
σ = P
Figure: (a) A cylinder with uniform dipole distribution. (b) Equivalent
charge distribution.

22. Calculation of the charge induced on the surface of a
dielectric
In general if the polarization vector makes an angle θ with n̂ , the
outward normal vector of the surface, the surface charge density
would be
σb = P.n̂
Figure: A Polarized cylinder

23. Calculation of the charged induced within the dielectric
Figure: a diverging P results in a pileup of negative charge
If the polarization is nonuniform, we get accumulation of bound
charge within the material, as well as on the surface.
Z
ρbdτ = −
I
P.da = −
Z
v
∇.Pdτ
ρb = −∇.P

24. The electric displacement
The effect of polarization is to produce accumulations of (bound)
charge, ρb = −∇.P within the dielectric and σb = P.n̂ on the
surface.
Within the dielectric, the total electric charge density can be
written:
ρ = ρb + ρf
and the Gauss’s law reads
ϵ0∇.E = ρ = ρb + ρf = −∇.P + ρf
∇.(ϵ0E + P) = ρf
Where,
D = ϵ0E + P
D is called the electric displacement. In terms of D, Gauss’s law
reads
∇.D = ρf

25. Linear dielectrics
In linear dielectrics, the polarization is proportional to the electric
field, provided the electric field is not too strong:
P = ϵ0χeE
Where χe is called the electric susceptibility of the medium.
D = ϵ0E + P = ϵ0(1 + χe)E
D = ϵE
ϵ = ϵ0(1 + χe)
The constant ϵ is called the permittivity of the material.
The constant
ϵr =
ϵ
ϵ0
= 1 + χe
is called the dielectric constant of the material.

26. THE END.