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---

genius Physics……
...
e
...

• Due to this reason our clothes stick to our
body
...

(i) Glass (ii) Flannel (iii) Wool (iv) Silk (v) Hard Metal (vi)
Hard rubber (vii) Sealing wax (viii) Resin (ix) Sulphur

ELECTRIC CHARGE : Electric charge is
characteristic developed in particle of material
due to which it exert force on other such
particles
...


•

Charge cannot exist without material carrying
it

•

It is possible to develop the charge by rubbing
two solids having friction
...


•

Electrification due to friction is called frictional
electricity
...

There are two types of charges
...

• Similar charges repel each other,
• Opposite charges attract each other
...


•

Any particle has vast amount of charges
...


•

Inequality of charges give the material a net
charge which is equal to the difference of the
two type of charges
...
+vein one body and –ve in second
...

• The loss of electrons develops +ve charge
...
Transferring
lighter electron is easier
...

Charge and Mass relation
• Charge cannot exist without matter
...

• Hence if there is charge transfer, mass is also
transferred
...

Conductors, Insulators and Semiconductors
• Conductors : Material in which electrons can
move easily and freely
...
Metals, Tap water, human body
...
Hence Brass is a
conductor
...

The electron revolving around it is negatively
charged
...

With friction there is transfer of electrons,
hence net charge is developed in the particles
...


Insulator : Material in which charge cannot
move freely
...
Glass, pure water, plastic etc
...
Pradeep Kshetrapal

•

•

Electrons can be forced to move across an
insulator by applying strong force (called
electric field
...


dielectric strength
...

• Semiconductor : is a material which under
little stimulation (heat or Elect
...

Ex
...

• Superconductor : is that material which
presents no resistance to the movement of the
charge through it
...

Electrostatic Induction
• Phenomenon of polarization of charges in a
body, when a charged body is present near it,
is called electrostatic induction
...

•

Charging

by

Induction

Electrostatics

Basic properties of Electric charge
•
•
•

•

Additivity of Electric charges
Quantization of Electric charge
Conservation of Electric Charge

Additivity of Charges
...
Addition of positive and negative
charge makes Zero charge
Quantization of Electric charge
• Principle: Electric charge is not a continuous
quantity, but is an integral multiple of
minimum charge ( e)
...

• The material which is transferred during
electrification is an electron, in integral
numbers
...

• Charge on an electron (-e) and charge on a
proton (+e) are equal and opposite, and are
the minimum
...
6 x 10-19 coulomb
...
6 x 10-19 C
One proton has charge + 1
...
In this example, a negatively
charged rod pushes some of the negatively
charged electrons to the far side of a nearby
copper sphere because like charges repel each
other
...


Charge on a body Q is given by
Q = + ne
Where n is a whole number 1,2,3…
...
6 x 10-19

•

since e is smallest value of charge, it is called
Elementary Charge or Fundamental charge

•

( Quarks : In new theories of proton and
neutrons, a required constituent particles
called Quarks which carry charges +(1/3)e or
+(2/3)e
...


genius Physics

Copyright-- Pradeep Kshetrapal

2011

Notes only for class students
...
Pradeep Kshetrapal

•

But because free quarks do not exist and their
sum is always an integral number, it does not
violet the quantization rules
...


•
•

Conservation of Charges
Like conservation of energy, and Momentum,
the electric charges also follow the rules of
conservation
...
Isolated (Individual) Electric charge can neither
be created nor destroyed, it can only be
transferred
...
Charges in pair can be created or destroyed
...

At Nuclear level : Decay of U-238
238

U

234

Th + 4 He

(Radio active decay)

Atomic number Z of radioactive material U-238 is 92
...
Thorium
has Z= 90, hence charge is 90e, alpha particles have
charge 2e
...
(a) Annihilation (destruction in pair)
In a nuclear process an electron -e and its antiparticle
positron +e undergo annihilation process in which they
transform into two gamma rays (high energy light)
e- + e + y + y

If two charges q1 and q2 are placed at distance r then,

where c is a constant
...

Accordingly value of c is 9 x 109 Newton x m2/coul2

Notes only for class students
...
Pradeep Kshetrapal

e0 is permittivity of free space or vacuum and its value
is e0 = 8
...
For vacuum e = e0

Permittivity, Relative Permittivity and Dielectric
Constant
Permittivity is a measure of the property of the
medium surrounding electric charge which determine
the forces between the charges
...

For water, permittivity is 80 times then that of vacuum,
hence force between two charges in water will be 1/80
time force in vacuum (or air
...
r
...
permittivity of vacuum
or air
...
I
...

Therefore coulomb is defined in it’s terms as
under:
Coulomb is that quantity of charge which
passes across any section of a conductor per
second when current of one ampere flows
through it, i
...

1 coulomb=1 Ampere x 1 sec

•

•

In cgs electrostatic system, the unit of charge is
called as STATECOULUMB or esu of charge
...

In cgs electromagnetic system, the unit of charge is
called ABCOULOMB or emu of charge
1 Coulomb = 3 x 109 statcoulomb
= 1/10 abcoulomb
\ 1 emu = 3x1010 esu of charge
Vector form of Coulumbs’ Law
Equation of Coulumbs force showing magnitude as
well as direction is called Vector form of coulumbs’
law
...
I
...


•

One coulomb is defined as that charge, which,
when placed at a distance of 1 m in air or
vacuum from an equal and similar charge,
repel it with a force of 9 x 109 Newton
• Charge on one electron is 1
...

Hence
• One coulomb is equivalent to a charge of 6
...
1
...
2

21

12

21

= -

12

Electrostatic Force between two point charges in terms
of their position vectors
...
Let there be two point charges q1 and q2 at points A
& B in vacume
...


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Page 4

genius Physics……
...
Then
AB=

...

F1 = F11 + F12 + F13 + …… + F1n
The vector sum is obtained as usual by parallelogram law of
vectors
...


Y

12

A

B

q1

q2

21

X
(ii) According to Coulumb’s law, the Force
by q2 is given by :

12 =

21

12

exerted on q1

where

vector pointing from q2 to q1
...
Identical Properties :
 Both the forces are central forces, i
...
, they act along
the line joining the centers of two charged bodies
...
e
...

 Both the forces are effective even in free space
...
Non identical properties :
a
...

b
...

c
...
How many electrons must be removed from the sphere to give it
a charge of +2 μC
...
How much is this change?
2
...
A third sphere of same size but
uncharged is brought in contact with the first, then brought in
contact with the second and finally removed from both
...
A central particle of charge –q is surrounded by two circular
rings of charged particles, of radii r and R, such that R > r
...


Qn
...

Ans : Fe=
Fg=G

= 9x10

9

Newton

-11

=6
...
26 x 1039
Principle of Superposition of Charges :
genius Physics

Copyright-- Pradeep Kshetrapal

Notes only for class students
...
Pradeep Kshetrapal

4
...
0 x 10-6 are fixed at three corners
of an equilateral triangle of side 5 cm
...

5
...
A distance h directly
beneath each of these spheres is a fixed sphere with positive
charge Q
...
Find the distance x when the rod is horizontal
and balanced
...
A charge q is placed at the center of the line joining two
equal charges Q
...

7
...
where, in the line joining the two charges, a
proton be placed so that it is in equilibrium (the net force is
zero)
...
What are the horizontal and vertical components of the
net electrostatic force on the charged particle in the lower
7
left corner of the square if q = 1
...
0 cm?
+q

a

a

a

9
...
Assume that θ is so
small that tan θ can be replaced
by sin θ; show that,
for equilibrium,

10
...

Find (a) the magnitude of q, assuming that
the charge on each balloon is at its centre
and (b) the volume of each balloon
...
29 kg m
and the density of helium in the balloon
m-3
is= 0
...
Neglect the weight of the
-7
-3
unfilled balloons
...
5 x 10
V = 2
...
8
...
At the left and right ends of the
rod are attached small conducting spheres with positive
Copyright-- Pradeep Kshetrapal

m

3

11
...
The strings make an angle of 30 with each other
...
What is the dielectric constant of the liquid? The
-3
density of the material of the sphere is 1600 kg m
Ans : K = 2

L

x

genius Physics

(a) What is the magnitude of the net electrostatic force exerted on
Cl- ion by the eight Cs+ ions
...

How much will be the force on chlorine ion in that case?

a

+2q

X=(

-q

9
...
Edge length is 0
...


Notes only

12
...
Two beads of equal
masses m each and carrying charges q1 q2 are connected by a
cord of length l and can slide without friction on the wires
...

If the cord is now cut what are
the value of the charges for
which the beads continue
for class students
...
Pradeep Kshetrapal

Electrostatics

2011

charge
...
Linear distribution:
when charge is evenly
distributed over a length
...
Which has relation

ELECTRIC FIELD
ELECTRIC FIELD-is the environment created by an
electric charge (source charge) in the space around it,
such that if any other electric charges(test charges)is
present in this space, it will come to know of its
presence and exert a force on it
...


Direction of force F is in direction of electric field E

Where Q is charge given to a surface of area ‘S’
...
1and 3 : Intensity of electric field due to Source
charge Q is

The surface charge density is ‘σ‘
given by

3-volumetric distribution: charge is
evenly distributed throughout the body
having volume ’V’Volumetric charge density is ‘ρ‘
GENERAL DISTRIBUTION OF ELECTRIC FIELD
DUE TO DIFFERENT DISTRIBUTION OF
CHARGES

By coloumb’s law we know that in similar situation
if q=1 then

1-Due to point change Q
2-E due to linear distribution of electric charge

Relation in F, E and Test charge q is

DISTRIBUTION OF CHARGE
Electric charge on a body may be concentrated at a
point, then it is called a ‘point charge’
...


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genius Physics……
...

example: molecule of electrolytic compounds
...

2-CO2 & CH4 are non-polar because centers of –ve &
+ve charges co-incide and there is no distance between
them
...
field a distance
is created between +ve & -ve charge: it become polar
...
Dipole

IF R>>L THE,

2

E=

ON EQUATORIAL LINE (TRANSVERAL LINE)

moment
...
(each charge is at a distance L from ‘center’ of
dipole)
Dipole moment = q x 2 is a vector quantity it has
magnitude p=2qL
And its direction is along line from –q to +q
...
One along
equatorial line and other in axial directions which are
the Esinθ and normal direction E cosθ
...


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genius Physics……
...


: net electric field E = 2E cosθ
E(net) = 2Ecosθ

E=

3/2

IF R>>L Then,

E=

The direction is opposite to that of P
-----------------------------------------------------------------------Electric Field at equatorial line is half of the field on
axial line in strength and opposite in direction
...

These are imaginary lines which give visual idea of
Electric field, its magnitude, and direction
...


Electric Field at A is stronger than field at B
...


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genius Physics……
...
start from positive charge and end at negative
...
Electric Lines of forces are imaginary but Electric
field they represent is real
...
The tangent drawn at any point on the line of force
gives the direction of force acting on a positive charge
at that point
...
In SI system, the number of electric lines originating
or terminating on charge q is q/ε
...
Two lines of force never cross each other, because if
they do so then at the point of intersection, intensity
will have two directions which is absurd
...
Electric Lines of force can never be a closed loop
since they do not start and end at the same point
...
The electric line of force do not pass through a
conductor as electric field inside a conductor is zero
...
Lines of force have tendency to contract
longitudinally like a stretched string, producing
attraction between opposite charges and edge effect
...
Electric Lines of force start and end Normal to the
surface of conductor
...
Crowded lines represent strong field while distant
lines represent weak field
...
Non-straight or non- parallel
represent non-uniform field
...


Electrostatics

2011

Lines of force due to Two positive charges

Elect field lines due to straight line distribution :
And Electric field lines due to very large sheet of
charge are shown in the previous page
...
Positive, in the direction of
field and negative, opposite to direction of field
...

1
...


Two equal and opposite charges :

Torque on dipole: A couple of force is acting on the
body of dipole system at different points, the forces
are equal and opposite in uniform field
...

Therefore dipole is capable of rotation in a uniform
electric field
...


= qEx2Lsinθ = 2qL E Sinθ = PESinθ
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genius Physics……
...
Direction of torque is normal to the plane containing
dipole moment P and electric field E and is governed
by right hand screw rule
...
If Dipole is parallel to E the torque is Zero
...
Torque is maximum when Dipole is perpendicular to
E and that torque is PE

= pE

3
...
If the dipole is rotated through 1800 from the
direction of the field, then work done will be :
W = pE

= 2 pE

4
...

If E is 1 N/C then P=T
...

5
...
But when it reach along the direction
of E the torque become zero
...

6
...
At the same time due to couple of
forces acting, a torque will also be acting on it
...
If a dipole is placed in a uniform electric field
experience a torque
...
If an Electric dipole
with moment P is placed in electric field E making an
angle α, then torque acting on it at that instant is
τ = PESinα
2
...
dα
Then work done for rotating it through an angle θ from
equilibrium position of angle 0 is :W=
genius Physics

= PE

Potential Energy of a dipole kept in Electric field :
1
...
e
...

1
...
Work done on +q = +(Work done up to B)
Adding the two
Total work done = Work done on –q from B to A
= Force x displacement
= -qE x 2L = - 2qLE
=- P
...
E
If P and E are inclined at angle θ to each other then
magnitude of this Potential Energy is
U = - P E Cos θ

Electric – Potential
(1)

θ

Copyright-- Pradeep Kshetrapal

Notes only for class students
...
If a unit
charge is placed at that location it has
potential energy (due to work done on
its placement at that location)
...

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genius Physics……
...

A
B

VA

Potential at A --------------------------VA

Energy with Q at B

2011

Hence applied force F = Work

done

in

moving

distance

dr

is

dw = Total work done in bringing the charge from
distance
to distance r is

qV

Energy with q at A is

Electrostatics

W =

q VA
is

= -

q VB

Difference of Energy UA – UB = q (VA – VB)
= -

=

Using work energy theorem
...


W/q =

OR

V=

Potential V = 0 , Inside
Where Q is source charge, r is distance & V r is

Then VA = W / q
This equation gives definition of potential V at
point A as under :“Potential of a point in electric field is the work
done in bringing a unit charge from infinity (Zero
potential) to that point, without any acceleration
...

Basically

V r is also a “potential difference”

between potential of this point P and Potential at
(i
...
, 0)
...
Point P is at distance ‘r”
from it
...

dr
P
Q

A test charge q is moved against E for a small distance
dr
...


for

a

Or, dw / q = - E dr
Or,
dv = - E
...
–
ve sign show that direction of E is opposite to
direction of dv
...
e
...


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genius Physics……
...
dr
Or V = E
...


Or

2011

2)

Then V =

P
4 0 r 2

At a point on equatorial line

- q & + q are placed at A & B
...

Principle of super position:1)
Potential at a point due to different charges
is Algebric sum of potentials due to all individual
charges
...
Therefore +ve & -ve potential are equal Hence net
potential is zero
...
”
iii) Potential due dipole at any general point
...


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Page 13

genius Physics……
...
The electric field is
directed normal to surface i
...
, Radially outward
...

Draw normal from A & B on PO
Hence potential at distance r is V =
PB

PN = PO – ON = r – L Cos  ------------ (i)

PA

PM = PO + OM = r + L Cos  ----------- (ii)

Potential on the surface of shell V =

V+q =

Q
Q
=
4 0 PB
4 0 (r  L cos )

V-q =

Q
Q
=
4 0 (r  L cos  )
4 0 PA

Total

V

=

V+q +

Q 
1
1




4 0  r  LCos  r  LCos  
=

V-q

Q
4 0 R

Inside shell Electric field is Zero
...
e
...

Hence potential
inside a spherical shell is same as on the surface and it
is same at every point
...

4 0 R
=
Relation of V & r for spherical shell

Q  r  L cos   r  LCos  


4 0 
r 2  L2Cos 2


=

Q
4 0 r

Q X 2 LCos 
4 0 (r 2  L2Cos )

Or

V=

PCos
4 0 ( r 2  L2Cos )

In case of non-conducting sphere of charge
...

r

Q

If r > > L
Then,

P

Or,

V =

PCos
4 0 r 2

v

0,0
genius Physics

Copyright-- Pradeep Kshetrapal

Notes only for class students
...
Pradeep Kshetrapal

A body of potential v’ is placed inside cavity of shell
with potential V then potential of the body become
V+v’

Electrostatics

ii)

V2

–

2011

V1

= dv

= - E

Cos 

E
V2

V+v ’

v’


...
P
...
Equipotential Surface
A real or imaginary surface in an electric field which
has same potential at very point is an equipotential
surface or simply, an equipotential
...

4 0 R
Electric
lines of
force and
equipotent
ial surface
are at right
angle to
each other
...

Distribution of charge on uneven surface: - charge
density is more on the surface which is pointed, or has
smaller radius
...
Electric field set up will be very
strong
...


Gauss's Law
Electric Flux

Proof:Suppose E is
not at right angle to equipotential surface, and makes
angle  with it
...
But
we find that charges are in equilibrium
...
e
...
How much
air comes through the window depends upon the speed
of the air, the direction of the air, and the area of the
window
...

We will define the electric flux
that is perpendicular to an area as

for an electric field

=EA

E Cos  = 0 ;
since E = 0, therefore Cos  = 0 or ∠  = 900
Hence E is always at right angle to equip
...


genius Physics

Copyright-- Pradeep Kshetrapal

Notes only for class students
...
Pradeep Kshetrapal

Electrostatics

2011

For a curved surface, that will not be the case
...


If the electric field E is not perpendicular to the area,
we will have to modify this to account for that
...
The "effective area" is A cos
or the component of the velocity perpendicular to the
window is v cos
...


A cos

and

or

A

To find the flux through all of a closed surface, we need
to sum up all these contributions of
over the entire
surface,
Cosθ
We will consider flux as positive if the electric field E
goes from the inside to the outside of the surface and
we will consider flux as negative if the electric field E
goes from the outside to the inside of the surface
...


Remembering the "dot product" or the "scalar product",
we can also write this as
=E A
where E is the electric field and A is a vector equal to
the area A and in a direction perpendicular to that
area
...
In that case, we could also write the electric flux
across an area as
=E nA
Both forms say the same thing
...


Gauss’s Law : Total electric flux though a closed
surface is 1/ε₀ times the charge enclosed in the
surface
...


genius Physics

Copyright-- Pradeep Kshetrapal

Notes only for class students
...
Pradeep Kshetrapal

PROOF : Let’s consider an hypothetical spherical
surface having charge q placed at its centre
...


Electrostatics

2011

Consider a Gaussian Surface in the shape of a cylinder
having axis along conductor
...
Let its length be l
...


q

At the small area flux dφ =

+
+

=

plain

+

=

+
+
+
+
+
+
+
+
+
+

, Cos0=1)

ds (E=

=
is 4πr2
...


Or, Φ = q / ε₀

Now

=

P

curved

E

r

for curved surface

+

for 2

plane surfaces
...
(Hence proved)

=
+
=E
for curved surface ( E is uniform)
= E2πrl (
2πrl, for cylindrical curved surface)

Application of Gauss’s Law
The charge enclosed within Guassian surface =λl
To calculate Electric Field due to different charge
distributions
...


According to Gauss theorem :
Putting values : E2πrl = λl / ε₀
Or,

Guassian Surface : It is an imaginary surface in the
electric field which is
1
...
Surface is Symmetrical about the charges in it
3
...
We have to
find Electric field on a point P at normal distance r
...
There is a point P at normal distance r
...

Electric field E is normal to the surface containing
charge hence it is normal to the plain surface of
cylinder and parallel to curved surface
...


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Page 17

genius Physics……
...
(for spherical shell
= 4πr2 )

++++

curved

Electrostatics

E=

for curved surface +
for 2
plane surfaces
...
A
Or, q = σπa2
Applying Gauss’s Law :

Case 2
...

In above formula when r decrease to R the electric
field increase
...
i
...
In this case if
complete charge q is placed at the centre of shell the
electric field is same
...


Electric Field due to charge distributed over a
spherical shell :-

Case 3
...
Or ‘r’ R
Charged Shell

+
+

+
+
+
+
+
+

+
+
+
+
+
+
+
+

+
+

R

+

+
+
+
+
+

+

r
P

E

R

+
+
+
+ Gaussian Surface
+
+

r

+

Then charge contained inside Gaussian surface is Zero
...

As ds is not zero then E = 0

+

The spherical shell or spherical conductor has total
charge q, surface charge density σ , radius R
...

Case 1
...

Let’s assume a Gaussian surface, which is a concentric
sphere of radius r and P lies on its surface
...

Hence it is radially outward
...
e
...


Copyright-- Pradeep Kshetrapal

+

Let’s consider +a Gaussian surface, a concentric
spherical shell of radius r passing through P
...

The electric field inside conductor is Zero
...

Variation of E with r ( distance from centre)

Notes only for class students
...
Pradeep Kshetrapal

Electrostatics

2011

Putting values E x 4πr2 = ρ πr / ε₀
E=
Surface
E=0

It shows that inside a sphere of charge, the electric
field is directly proportional to distance from centre
...
When+P is Out + + + sphere
...
When + + + +P is + + the surface of shell: Same
+++++++++
as in case of shell
...

+ + + is + + +
Consider Gaussian surface, a concentric spherical shell
++++++++
of radius r, such that point P lies on the surface
...

Now
=
for complete area of Gaussian surface
=
= E
(E is uniform)
2
= E x 4πr
...

= ρ πr3 (where ρ is the charge per unit volume
...
Charges of opposite nature :+
1
2+
+
+ E =+
+ E1 = +
E1 = 1
+
+
+
+
+
+
E2 = +
E2 = + E2 = +
+
+
E= -E1 + E2= 0
E=+E1+E2
E= +E1 - E2= 0
+
=+
+
+
+
+
Equipotential Surface :
+
Energyof a charged particle in terms of potential:+
E1 = -

= q / ε₀

Copyright-- Pradeep Kshetrapal

Notes only for class students
...
Pradeep Kshetrapal

Electrostatics

2011

Work required to bring a charge q at a point of
potential V is W = qV
...

Potential energy of charge q at potential V is U = qV
Electron-Volt : By relation Work/energy = qV,
smallest unit of work/energy is Electron Volt
...

1 eV = 1
...
Due to
induction, they induce large number of opposite
charges
...
Finally the dielectric strength breaksdown and there is fast flow of charges
...


System of Two charges :

A
B
q1 -------- r ----------------q2
Potential due to q1 at B is potential at distance r :
V =
Potential Energy of system U =

(ii)

System of three charges

We make different pairs and calculate energy as under
U=
+
+

(iii)

System of Four charges

Four charges make six pairs : Potential Energy U=

+

+

+

+

+

The energy is contained in the system and not by
any one member
...

Distribution of charge on irregular shaped conductors :
Van-de-Graff generator

Potential at each point is equal
...

Charge is distributed unevenly
...

Therefore charge density is always more on the
corners
...

Principles : The following principles are involved in the
device
...
Charge on a conductor always move to and stay on
the outer surface
...
Pointed Corners conduct charges very effectively
...


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genius Physics……
...
If charge q is given to a body, its potential increases
by relation V=
4
...

A long belt of insulating material like silk rubber or
rayon moves around two pulleys, driven by a motor
...

Lower one is spray comb and the upper Collecting
Comb
...

There is a discharge tube
...
Target is
placed at the other end connected to earth
...

Working : The spray comb is given a positive potential
4
(
Volt) w
...
t
...

Due to sharp points there is spray of charge on belt
...
When the
charges reach near upper comb, due to induction and
corona discharge the charge on belt is transferred to
comb
...

Since charge always stay at the outer surface, it moves
to outer surface and the inner surface again become
without any charge, ready to receive fresh charge
again
...
This potential is

Relation between Equipotential surfaces and E-Lines

distributed all over and inside the shell
...
There new
potential is slightly higher than shell, therefore charges
move from belt to comb to shell
...
This process keeps on repeating and V
increase to a very high value, that is break-down
voltage of compressed nitrogen 107volt
...
Its energy
increases by relation U = qV
...
Hence this ion gets accelerated
and hits the target with very high energy
...


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genius Physics……
...

Any conductor can store charge to some extent
...
When
charge is given to a conductor its potential increases
...

When due to increasing charge the potential increase
to such extent that air touching the conductor starts
getting ionized and hence charge gets leaked
...
This is limit of charging a conductor
...

Principle : Principle of a parallel plate capacitor is
that an uncharged plate brought bear a charged
plate decrease the potential of charged plate and
hence its capacitance (C = ) increase
...


take more charge
...
This
arrangement of two parallel plates is called parallel
plate capacitor
...
Therefore ,
Capacitance of a conductor is equal to the charge
which can change its potential by one volt
...

One farad is capacitance of such a conductor whose
potential increase by one volt when charge of one
coulomb is given to it
...
The practical smaller
units are
i
...
(used in electrical circuits)
Ii Pieco farad ( pF) = 10-12 used in electronics circuits
Expression for capacitance of a spherical conductor :
If charge q is given to a spherical conductor of radius r,
its potential rise by V =
Therefore capacitance C =

Or for a sphere

= q/

=

C=

The capacitor depends only on the radius or size
of the conductor
...


is

+

+
is kept at a distance ‘d’
...
Here
d
+
q = σA
and E=
+
The Potential difference between plates is given by
V = Ed =
d
+
Now C = =
=
+
Of area ‘A’
...
e
...

The other examples of parallel plate capacitors is
Cylindrical capacitor C =
and Spherical capacitor
...


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genius Physics……
...
1
...
Parallel
...


Parallel combination :

then equivalent capacitance Ce =

If capacitors are connected in such a way that there
are many paths to go from one point to other
...


Here three capacitors are connected in series and are
connected across a battery of P
...
‘V’
...
Due to induction it attract –q on the
opposite plate
...
Same action is repeated to all
the capacitors and in this way all capacitors get q
charge
...

The Potential Difference V of battery is sum of
potentials across all capacitors
...
D
...

The potential difference across each capacitor is equal
and it is same as P
...
across Battery
...
The total charge
q = q1 + q2 + q3
Each capacitor has charge
q1=c1v1, q2=c2v2, q3=c3v3
Equivalent Capacitance : We know that
q = q1 + q2 + q3
divide by v
= + +
or, C = c1+c2+c3
The equivalent capacitance in parallel increases, and it
is more than largest in parallel
...

In series q is same
...
e
...

For 2 capacitor system C =

, and v1 =


...

Charge distribution :

q1=c1v,

c
...


In 2 capacitor system charge on one capacitor
q1 =

...


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genius Physics……
...
Hence a new charge
has to be sent by applying force and doing work on it
...

At any instant work done dw = V
...
dq
Therefore work done in charging
the capacitor from charge 0 to q

W=

Electrostatics

2011

Energy in combination :

(

(

)²

Hence Loss in energy : E1 – E2
= { c1v12 +
=

c2v22 } – {

(

(

)²}

) (v₁ − v₂)²

(

=

=
=
This work done convert into electrical Potential
Energy stored in the capacitor U =

=

It is a positive number which confirm that there is
loss of energy in transfer of charges
...


Wheatstone bridge in combination of capacitors :

Energy per unit volume u = cv2/V =
Or, energy density u

Five capacitors joined in following manner is called
wheatstone bridge connection
...

Common Potential : A capacitor of capacitance c1
and potential v1 is connected to another capacitor
of capacitance c2 and potential v2
...
The charge flow from higher
potential to lower potential and in this process it
looses some energy as charge has to do some
work while passing through connecting wire
...

Expression for energy lost : In the above two
capacitors the energy contained in the two before
connection, E1 = c1v12 + c2v22
...
In such case
the potential at point Q and S are equal
...
In this way it is not participating
in storage of charges
...
Calculations are done for c1
c2, c3 and c4 only
...
They do not have
free charged particles like conductors have
...

i
...
Example HCl, H2O, They have their own
dipole moment
...


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genius Physics……
...
Example CO2 , C6H6
...

In both cases, when a dielectric slab is exposed to an
electric field, the two charges experience force in opposite
directions
...

surface charge density σp and not the volumetric charge
density
...
Then net electric field E is given by E = E o - Ep
...

ii
...


Clearly electric field inside a dielectric is

E=


...
i
...
p Eo
or p = αε₀E₀
here α is a constant called atomic / molecular polarizability
...

This polarization is a vector quantity and is related to
resultant electric field E as under :
= χeεE
Where χe is a constant called electric susceptibility of the
dielectric
...


= E - Ep =

=
or (

=σ
or

The quantity

=σ

is called electric displacement in dielectric
...


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