Electrostatics Class 12- Part 4

ELECTROSTATICS - IV
- Capacitance and Van de Graaff
Generator1. Behaviour of Conductors in Electrostatic Field
2. Electrical Capacitance
3. Principle of Capacitance
4. Capacitance of a Parallel Plate Capacitor
5. Series and Parallel Combination of Capacitors
6. Energy Stored in a Capacitor and Energy Density
7. Energy Stored in Series and Parallel Combination of Capacitors
8. Loss of Energy on Sharing Charges Between Two Capacitors
9. Polar and Non-polar Molecules
10.Polarization of a Dielectric
11.Polarizing Vector and Dielectric Strength
12.Parallel Plate Capacitor with a Dielectric Slab
13.Van de Graaff Generator
Created by C. Mani, Principal, K V No.1, AFS, Jalahalli West, Bangalore

Behaviour of Conductors in the Electrostatic Field:
1. Net electric field intensity in the interior of a
conductor is zero.
When a conductor is placed in an
electrostatic field, the charges (free
electrons) drift towards the positive plate
leaving the + ve core behind. At an
equilibrium, the electric field due to the
polarisation becomes equal to the applied
field. So, the net electrostatic field inside
the conductor is zero.
E0
Enet = 0
EP
2. Electric field just outside the charged
conductor is perpendicular to the surface of
the conductor.
Suppose the electric field is acting at an
angle other than 90°, then there will be a
component E cos θ acting along the tangent
at that point to the surface which will tend to
accelerate the charge on the surface leading
to ‘surface current’. But there is no surface
current in electrostatics. So, θ = 90° and
cos 90° = 0.
n
E
θ
E cos θ
+ q•
NOT POSSIBLE

3. Net charge in the interior of a conductor is zero.
The charges are temporarily separated. The total
charge of the system is zero.
E . dS =
S
ΦE =
q
ε0
Since E = 0 in the interior of the conductor,
therefore q = 0.
4. Charge always resides on the surface of a
conductor.
Suppose a conductor is given some excess
charge q. Construct a Gaussian surface just
inside the conductor.
therefore q = 0 inside the conductor. q = 0
q q
5. Electric potential is constant for the entire
conductor.
dV = - E . dr
therefore dV = 0. i.e. V = constant

6. Surface charge distribution may be different
at different points.
σ =
q
S
σmax
σmin
Every conductor is an equipotential volume
(three- dimensional) rather than just an
equipotential surface (two- dimensional).
Electrical Capacitance:
The measure of the ability of a conductor to store charges is known as
capacitance or capacity (old name).
q α V or q = C V or C =
q
V
If V = 1 volt, then C = q
Capacitance of a conductor is defined as the charge required to raise its
potential through one unit.
SI Unit of capacitance is ‘farad’ (F). Symbol of capacitance:
Capacitance is said to be 1 farad when 1 coulomb of charge raises the
potential of conductor by 1 volt.
Since 1 coulomb is the big amount of charge, the capacitance will be usually
in the range of milli farad, micro farad, nano farad or pico farad.

Capacitance of an Isolated Spherical Conductor:
O •
r
+q
Let a charge q be given to the sphere which
is assumed to be concentrated at the centre.
Potential at any point on the surface is
V =
q
4πε0 r
C =
q
V
C = 4πε0
r
1. Capacitance of a spherical conductor is directly proportional to its radius.
2. The above equation is true for conducting spheres, hollow or solid.
3. IF the sphere is in a medium, then C = 4πε0εr r.
4. Capacitance of the earth is 711 μF.

Principle of Capacitance:
E
A B
A
Step 1: Plate A is positively charged and B is neutral.
Step 2: When a neutral plate B is brought near A,
charges are induced on B such that the side near A is
negative and the other side is positive.
The potential of the system of A and B in step 1 and 2
remains the same because the potential due to positive
and negative charges on B cancel out.
Step 3: When the farther side of B is earthed the
positive charges on B get neutralised and B is left only
with negative charges.
Now, the net potential of the system decreases due to
the sum of positive potential on A and negative potential
on B.
To increase the potential to the same value as was in
step 2, an additional amount of charges can be given to
plate A.
This means, the capacity of storing charges on A
increases.
The system so formed is called a ‘capacitor’.
Potential = V
Potential = V
Potential
decreases to v
B

Capacitance of Parallel Plate Capacitor:
Parallel plate capacitor is an arrangement of two
parallel conducting plates of equal area
separated by air medium or any other insulating
medium such as paper, mica, glass, wood,
ceramic, etc.
A
σ
d
A
σ E
V = E d
σ
ε0
= d
or
q d
V =
A ε0
But
d
C =
A ε0
C =
q
V
If the space between the plates is filled with dielectric medium of
relative permittivity εr, then
d
C =
A ε0 εr
Capacitance of a parallel plate capacitor is
(i) directly proportional to the area of the plates and
(ii) inversely proportional to the distance of separation between them.

Series Combination of Capacitors:
V1 V2 V3
V
C1 C2 C3
In series combination,
i) Charge is same in each capacitor
ii) Potential is distributed in inverse
proportion to capacitances
i.e. V = V1 + V2 + V3
But
q
V1 =
C1
V2 =
C2
q
V3 =
C3
q
, and
q
V =
C
,
(where C is the equivalent capacitance or
effective capacitance or net capacitance or
total capacitance)
q
= C1
+
C2
q
+
C3
qq
C
or
The reciprocal of the effective capacitance is the sum of the reciprocals of the
individual capacitances.
Note: The effective capacitance in series combination is less than the least of
all the individual capacitances.
q q q
∑
i=1
n 1
Ci
1
C
=
1
= C1
+
C2
1
+
C3
11
C

Parallel Combination of Capacitors:
In parallel combination,
i) Potential is same across each capacitor
ii) Charge is distributed in direct proportion to
capacitances
i.e. q = q1 + q2 + q3
But , and,
(where C is the equivalent
capacitance)
or
The effective capacitance is the sum of the individual capacitances.
Note: The effective capacitance in parallel combination is larger than the
largest of all the individual capacitances.
q1 = C1 V q2 = C2 V q3 = C3 V q = C V
C V = C1V + C2 V + C3 V
∑
i=1
n
CiC =C = C1 + C2 + C3
V q1
C1
C2
C3
V
V
V
q2
q3

Energy Stored in a Capacitor:
V
The process of charging a capacitor is
equivalent to transferring charges from one
plate to the other of the capacitor.
The moment charging starts, there is a potential
difference between the plates. Therefore, to
transfer charges against the potential difference
some work is to be done. This work is stored as
electrostatic potential energy in the capacitor.
If dq be the charge transferred against the
potential difference V, then work done is
dU = dW = V dq
q
=
C
dq
The total work done ( energy) to transfer charge q is
U =
0
q
q
C
dq U =
q2
C
1
2
U =
1
2
C V2
U =
1
2
q Vor or or

Energy Density:
U =
1
2
C V2
d
C =
A ε0
V = E dand
U =
1
2
ε0 Ad E2
1
2
ε0 E2
=
U
Ad
1
2
ε0 E2
=U
But
or or
SI unit of energy density is J m-3
.
Energy density is generalised as energy per unit volume of the field.
Energy Stored in a Series Combination of Capacitors:
1
= C1
+
C2
1
+
C3
11
C
+
Cn
1
………. +
U =
q2
C
1
2
U =
1
2
q2
1
[ C1
+
C2
1
+
C3
1
+
Cn
1
………. + ]
U = U1 + U2 + U3 + ………. + Un
The total energy stored in the system is the sum of energy stored in the
individual capacitors.

Energy Stored in a Parallel Combination of Capacitors:
U = U1 + U2 + U3 + ………. + Un
The total energy stored in the system is the sum of energy stored in the
individual capacitors.
C = C1 + C2 + C3 + ……….. + Cn
U =
1
2
C V2
U =
1
2
V2 ( C1 + C2 + C3 + ……….. + Cn )
Loss of Energy on Sharing of Charges between the Capacitors in
Parallel:
Consider two capacitors of capacitances C1, C2, charges q1, q2 and
potentials V1,V2.
Total charge after sharing = Total charge before sharing
(C1 + C2) V = C1 V1 + C2 V2
V =
C1 V1 + C2 V2
C1 + C2

The total energy before sharing is
Ui =
1
2
C1 V1
2
1
2
C2 V2
2
+
The total energy after sharing is
Uf =
1
2
(C1 + C2) V2
Ui– Uf =
C1 C2 (V1 – V2)2
2 (C1 + C2)
Ui – Uf > 0 or Ui > Uf
Therefore, there is some loss of energy when two charged capacitors are
connected together.
The loss of energy appears as heat and the wire connecting the two capacitors
may become hot.

Polar Molecules:
A molecule in which the centre of positive charges does
not coincide with the centre of negative charges is called
a polar molecule.
Polar molecule does not have symmetrical shape.
Eg. H Cl, H2 O, N H3, C O2, alcohol, etc.
O
H H
105°
Effect of Electric Field on Polar Molecules:
E = 0 E
p = 0 p
In the absence of external electric
field, the permanent dipoles of the
molecules orient in random
directions and hence the net dipole
moment is zero.
When electric field is applied, the
dipoles orient themselves in a
regular fashion and hence dipole
moment is induced. Complete
allignment is not possible due to
thermal agitation.
p

Non - polar Molecules:
A molecule in which the centre of positive charges coincides with the centre of
negative charges is called a non-polar molecule.
Non-polar molecule has symmetrical shape.
Eg. N2 , C H4, O2, C6 H6, etc.
Effect of Electric Field on Non-polar Molecules:
E = 0 E
p = 0 p
In the absence of external
electric field, the effective
positive and negative centres
coincide and hence dipole is
not formed.
When electric field is applied, the positive
charges are pushed in the direction of electric
field and the electrons are pulled in the
direction opposite to the electric field. Due to
separation of effective centres of positive and
negative charges, dipole is formed.

Dielectrics:
Generally, a non-conducting medium or insulator is called a ‘dielectric’.
Precisely, the non-conducting materials in which induced charges are produced
on their faces on the application of electric fields are called dielectrics.
Eg. Air, H2, glass, mica, paraffin wax, transformer oil, etc.
Polarization of Dielectrics:
EpE0
When a non-polar dielectric slab is
subjected to an electric field, dipoles
are induced due to separation of
effective positive and negative centres.
E0 is the applied field and Ep is the
induced field in the dielectric.
The net field is EN = E0 – Ep
i.e. the field is reduced when a
dielectric slab is introduced.
The dielectric constant is given by
E0 - Ep
K =
E0
E = 0

Polarization Vector:
The polarization vector measures the degree of polarization of the dielectric. It
is defined as the dipole moment of the unit volume of the polarized dielectric.
If n is the number of atoms or molecules per unit volume of the dielectric, then
polarization vector is
P = n p
SI unit of polarization vector is C m-2
.
Dielectric Strength:
Dielectric strength is the maximum
value of the electric field intensity
that can be applied to the dielectric
without its electric break down.
Its SI unit is V m-1
.
Its practical unit is kV mm-1
.
Dielectric Dielectric strength (kV /
mm)
Vacuum ∞
Air 0.8 – 1
Porcelain 4 – 8
Pyrex 14
Paper 14 – 16
Rubber 21
Mica 160 – 200

Capacitance of Parallel Plate Capacitor with Dielectric Slab:
EpE0 EN = E0 - Ep
dt
V = E0 (d – t) + EN t
E
N
K =
E0
or EN =
K
E0
V = E0 (d – t) +
K
E0
t
V = E0 [(d – t) +
K
t
]
But E0 =
ε0
σ
=
ε0
qA
and C =
q
V
C =
A ε0
[(d – t) +
K
t
]
or C =
A ε0
d [1 –
K
t
]d
t
(1 - )
or C =
C0
[1 –
K
t
]d
t
(1 - )
C > C0. i.e. Capacitance increases with
introduction of dielectric slab.

If the dielectric slab occupies the whole space between the plates, i.e. t = d,
then
WITH DIELECTRIC SLAB
Physcial Quantity With Battery
disconnected
With Battery
connected
Charge Remains the same Increases (K C0 V0)
Capacitance Increases (K C0) Increases (K C0)
Electric Field Decreases
EN = E0 – Ep
Remains the same
Potential Difference Decreases Remains the same
Energy stored Remains the same Increases (K U0)
C0
K =
C
C = K C0
Dielectric Constant

Van de Graaff Generator:
T
D
C2
C1
P1
P2
M
S
I S
HVR
S – Large Copper sphere
C1, C2 – Combs with sharp points
P1, P2 – Pulleys to run belt
HVR – High Voltage Rectifier
M – Motor
IS – Insulating Stand
D – Gas Discharge Tube
T - Target

Principle:
Consider two charged conducting spherical shells such that one is
smaller and the other is larger. When the smaller one is kept inside the
larger one and connected together, charge from the smaller one is
transferred to larger shell irrespective of the higher potential of the larger
shell. i.e. The charge resides on the outer surface of the outer shell and
the potential of the outer shell increases considerably.
Sharp pointed surfaces of a conductor have large surface charge densities
and hence the electric field created by them is very high compared to the
dielectric strength of the dielectric (air).
Therefore air surrounding these conductors get ionized and the like
charges are repelled by the charged pointed conductors causing
discharging action known as Corona Discharge or Action of Points. The
sprayed charges moving with high speed cause electric wind.
Opposite charges are induced on the teeth of collecting comb (conductor)
and again opposite charges are induced on the outer surface of the
collecting sphere (Dome).

Construction:
Van de Graaff Generator consists of a large (about a few metres in
radius) copper spherical shell (S) supported on an insulating stand (IS)
which is of several metres high above the ground.
A belt made of insulating fabric (silk, rubber, etc.) is made to run over
the pulleys (P1, P2 ) operated by an electric motor (M) such that it ascends
on the side of the combs.
Comb (C1) near the lower pulley is connected to High Voltage Rectifier
(HVR) whose other end is earthed. Comb (C2) near the upper pulley is
connected to the sphere S through a conducting rod.
A tube (T) with the charged particles to be accelerated at its top and
the target at the bottom is placed as shown in the figure. The bottom end
of the tube is earthed for maintaining lower potential.
To avoid the leakage of charges from the sphere, the generator is
enclosed in the steel tank filled with air or nitrogen at very high pressure
(15 atmospheres).

Working:
Let the positive terminal of the High Voltage Rectifier (HVR) is
connected to the comb (C1). Due to action of points, electric wind is
caused and the positive charges are sprayed on to the belt (silk or
rubber). The belt made ascending by electric motor (EM) and pulley
(P1) carries these charges in the upward direction.
The comb (C2) is induced with the negative charges which are
carried by conduction to inner surface of the collecting sphere
(dome) S through a metallic wire which in turn induces positive
charges on the outer surface of the dome.
The comb (C2) being negatively charged causes electric wind by
spraying negative charges due to action of points which neutralize
the positive charges on the belt. Therefore the belt does not carry
any charge back while descending. (Thus the principle of
conservation of charge is obeyed.)
Contd..

The process continues for a longer time to store more and more
charges on the sphere and the potential of the sphere increases
considerably. When the charge on the sphere is very high, the
leakage of charges due to ionization of surrounding air also
increases.
Maximum potential occurs when the rate of charge carried in by
the belt is equal to the rate at which charge leaks from the shell
due to ionization of air.
Now, if the positively charged particles which are to be
accelerated are kept at the top of the tube T, they get accelerated
due to difference in potential (the lower end of the tube is
connected to the earth and hence at the lower potential) and are
made to hit the target for causing nuclear reactions, etc.

Uses:
Van de Graaff Generator is used to produce very high
potential difference (of the order of several million volts) for
accelerating charged particles.
The beam of accelerated charged particles are used to
trigger nuclear reactions.
The beam is used to break atoms for various experiments
in Physics.
In medicine, such beams are used to treat cancer.
It is used for research purposes.
END OF ELECTROSTATICS

Electrostatics Class 12- Part 4

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Electrostatics Class 12- Part 4