CHAPTER - 2 ELECTRIC POTENTIAL AND CAPACITORS2 theory phys… · Potential is a scalar quantity...

ELECTRIC POTENTIAL AND CAPACITORSPotential energy is defined only for conservative forces. In the space occupied by conservative forces every

point is associated with certain energy which is called the energy of position or potential energy. Potential

energy generally are of three types :

potential energy etc.

Change in potential energy : Change in potential energy between any two points is defined in the terms

of the work done by the associated conservative force in displacing the particle between these two points

without any change in kinetic energy.

∫−=−1

12

r

rUU

We can define a unique value of potential energy only by assigning some arbitrary value to a fixed point

called the reference point. Whenever and wherever possible, we take the reference point at infinite and

assume potential energy to be zero there,

∫∞−=r

rdFUrr

.

In case of conservative force (field) potential energy is equal to negative of work done in shifting the

body from reference position to given position.

This is why in shifting a particle in a conservative field (say gravitational or electric), if the particle

moves opposite to the field, work done by the field will be negative and so change in potential energy will

be positive i.e. potential energy will i

be positive and change in potential energy will be negative

We can define electric potential energy difference between two points as the work required to be done by

an external force in moving (without acceleration) a test charge q from one point to another in an electric

field.

∫∫ −==P

R

Eext

P

R

RP rd.Frd.FWrrrr

Potential energy difference U =∆

here work done by the electric field is negative i.e.

i) Work done to move a charge q depends only on initial and final point and not on the path taken. This

is the fundamental characteristics of conservative force.

ii) The actual value of potential energy is

energy that is significant.

If we take electric potential energy at infinity as zero ie R is at infinity we get

PPPP UUUUW =−=−= ∞∞ 0

21

CHAPTER - 2

ELECTRIC POTENTIAL AND CAPACITORSPotential energy is defined only for conservative forces. In the space occupied by conservative forces every


energy generally are of three types : Elastic potential energy, Electric potential energy and Gravitational

Change in potential energy between any two points is defined in the terms


without any change in kinetic energy.

−=2

1

.r

WrdFrr

……(i)

e value of potential energy only by assigning some arbitrary value to a fixed point


assume potential energy to be zero there, i.e. if take ∞=1r and rr =2 then from equation (i)

−= W


body from reference position to given position.



potential energy will increase. When the particle moves in the direction of field, work will

be positive and change in potential energy will be negative i.e. potential energy will decrease.

nergy difference between two points as the work required to be done by


RPRP WUU =−=

done by the electric field is negative i.e. - WRP

Work done to move a charge q depends only on initial and final point and not on the path taken. This

is the fundamental characteristics of conservative force.

The actual value of potential energy is not physically significant. It is only the difference in potential

If we take electric potential energy at infinity as zero ie R is at infinity we get W

ELECTRIC POTENTIAL AND CAPACITORS Potential energy is defined only for conservative forces. In the space occupied by conservative forces every


Elastic potential energy, Electric potential energy and Gravitational

Change in potential energy between any two points is defined in the terms


e value of potential energy only by assigning some arbitrary value to a fixed point


from equation (i)




ncrease. When the particle moves in the direction of field, work will

potential energy will decrease.

nergy difference between two points as the work required to be done by


Work done to move a charge q depends only on initial and final point and not on the path taken. This

not physically significant. It is only the difference in potential

RPRP UUW −=

potential energy of a charge Q at a point on an electric field is the work done by the external force in

bringing the charge q from infinity to that point.

Electrostatic Potential:

Electrostatic potential difference between two points (P&R) is defined as the work done by the external

force in bringing a unit positive charge form one point (R) to the other point (P)

Electrostatic potential difference = V

Potential is a scalar quantity unit is J/C or volt.

1 J/C = 1 Volt = V.

Smaller units: 1) micro volt ( =10

Electrostatic potential at a point is defined as the work done by an external force in bring a unit positive

charge from infinity to that point.

Electrostatic potential difference V∆

1. Give the physical concept of electric potential ?

The concept of potential plays an important role and it may be made clear as follows

Consider a region free of any electric charge. Imagine a small charge, +q being placed in the region. It

can be made to move about any where in the region without doing any work of electrical origin because

the charge does not experience any electric force.

Let a charge +Q be placed in the region before +q is introduced. Now, work has to be done to move the

charge +q from one point to another because it experiences electrostatic force of repulsion

presence of the charge Q in space endows every point in the medium with a property by virtue of which

work is to be done to move a charge up to this point. This property is called electrostatic potential.

Electrostatic potential of a body represents the degree of electrification of the body. It determines the

direction of flow of charge between two charged bodies placed in contact with each other. The charge

always flows from a body at higher potential to another body at lower p

as soon as the potentials of the two bodies become equal

2.Define electrostatic potential at a point and electrostatic potential difference between two points?

Electrostatic potential at a point is defined as the work

is taken is taken from infinity up to the point.

Electrostatic potential different between 2 points is the work done per unit charge when a positive test

charge is taken between the points.

Potential due to a point charge

Potential at a point P due to a charge Q is defined as the work done in taking a unit +1C charge from

infinity up to that point. Work done in moving through a small distance

22

Q at a point on an electric field is the work done by the external force in

bringing the charge q from infinity to that point.


force in bringing a unit positive charge form one point (R) to the other point (P)

q

UUVV RP

RP

−=−

Potential is a scalar quantity unit is J/C or volt. D. F of potential = ML2 T-3I -1.

=10-6

V 2) milli volt (m V) =10-3

V


q

W

eargch

workV

∆∆

==

. Give the physical concept of electric potential ?

The concept of potential plays an important role and it may be made clear as follows



the charge does not experience any electric force.


charge +q from one point to another because it experiences electrostatic force of repulsion



l of a body represents the degree of electrification of the body. It determines the


always flows from a body at higher potential to another body at lower potential. The flow of charge stops

as soon as the potentials of the two bodies become equal

.Define electrostatic potential at a point and electrostatic potential difference between two points?

Electrostatic potential at a point is defined as the work done per unit charge when a unit +ve test charge

is taken is taken from infinity up to the point.



to that point. Work done in moving through a small distance r′∆ is

Q at a point on an electric field is the work done by the external force in


.


The concept of potential plays an important role and it may be made clear as follows :




charge +q from one point to another because it experiences electrostatic force of repulsion. Thus the



l of a body represents the degree of electrification of the body. It determines the


otential. The flow of charge stops

.Define electrostatic potential at a point and electrostatic potential difference between two points?

done per unit charge when a unit +ve test charge



rdFW ′=∆2

0r4

Q

′πε= rd ′

∴ Total work done′

−= ∫∞

dr

QW

r

2

04πε

∴ Potential r

1V,

r

Q

4

1V

0

∝πε

= .The graph connecting V and r is a rectangular hyperbola

Potential due to an electric dipole

Consider an electric dipole of dipole moment

distance r from the centre of the dipole and making an angle

Potential due to a point charge q at a distance r is given by

and –q. Hence total potential at P

From the figure arr 1

rrr+= (triangular law of vector addition)

arr1rrr

−=

θ−+= cosra2arr 2221

+θ−=

2

2

221

r

acos

r

a21rr since r > > a

θ−= cosr

a21rr 22

1 1 cosr

a21rr

−=

2/1

1

cosr

a21

r

1

r

1−

θ−= using binomial expansion

θ+= cosr

a1

r

1

r

1

1

similarly arr2

rrr+=

θ++= cosra2arr 2222

23

∞

′

=′r

r

Qrd

04πε =

r4

Q

0πε

.The graph connecting V and r is a rectangular hyperbola

Consider an electric dipole of dipole moment a2qPrr

= . We want to find the potential at a point P at a

distance r from the centre of the dipole and making an angle θ as shown in figure.

Potential due to a point charge q at a distance r is given by r/q4

1V

0πε= . Here there are two charges +q

−=

24

1

10 r

q

r

qV

πε

(triangular law of vector addition)

since r > > a 2

2

r

a can be neglected

2/1

cos

θ

using binomial expansion

.The graph connecting V and r is a rectangular hyperbola

∝2

r

1E

. We want to find the potential at a point P at a

as shown in figure.

. Here there are two charges +q

+θ+=

2

222

2a

rcos

r

a21rr then

−= θcos11

2

1

r

a

rr

θ−−

θ+=− cosr

a1cos

r

a1

r

1

r

1

r

1

21

θ=− cosa2r

1

r

1

r

12

21

∴2

0

2

0

cos

4

1cos2

4 r

P

r

aqV

θπε

θπε

==

2

0 r

cosP

4

1V

θπε

= r̂.PcosPr

=θ

∴r̂.P

r4

1V

20

r

πε= ( r > > a)

Along the axial line π=θ ,0 4

Vπε

=

Along the equatorial line 2

π=θ , cos

(i) potential due to a dipole depends on r and

fixed, the points corresponding to P on the cone so generated will have the same potential as at P.

ii) due to a dipole 2

r

1Vα , due to a single charge

Potential due to a system of charges

nVVVV +++= .......21

+++=

nP

n

PP r

q

r

q

r

qV ....

4

1

2

2

1

1

0πε

Potential due to charged shell out the shell is given by

the shell and R its radius. Electric field inside the shell is zero this implies potential is a constant inside

the shell and equals value at its surface

Equipotential Surfaces

If surface has all points at the same potential it is called equi

24

2/1

cos2

11

2

1−

+= θr

a

rr

20 r

P1

πε

, cos 0,0 == Vθ

potential due to a dipole depends on r and θ. If you rotate the position vector


, due to a single charge r

1V ∝

Potential due to a system of charges

Potential due to charged shell out the shell is given by r

q

4

1V

0πε= ( )Rr≥ where q is the total charge on


the shell and equals value at its surface R

q

4

1V

0πε=

points at the same potential it is called equi potential surface.

. If you rotate the position vector rr

about Pr

keeping θ


where q is the total charge on


25

q1 q2

r12

For a single charge r

q

4

1V

0πε= . If V is constant r is constant. Equipotential surfaces of a single point

charge are concentric spherical surfaces centered at the charge.

For any charge configuration, equipotential surface through a point is normal to the electric field at that

point. Work done in moving a charge in an equipotential surface is zero.

3.Explain why +ve charge flows from higher potential to lower potential while -ve charge flows from

lower potential to higher potential?

Any system has a tendency to reduce the PE to q a minimum. The PE of a +vely charged particle will

decrease when it moves from higher potential to lower potential. The PE of a -ve charge will decrease

when it moves from lower potential to higher potential. This explains why a +ve charge flows from

higher potential to lower potential and –ve charge flows from lower potential to higher potential.

The mutual PE of a system of charges resides on the system of two charges and not with any one

single charge.

Relation between field and potential

A and B are two equi potential surfaces having potential V and V-δV respectively. Work done in taking a

unit positive charge from A to B is equal potential difference = ( ) VdVV −−

dV|E| −=δl

ld

dV|E|

−=

Thus electric field is equal to negative gradient of potential.

Notes

i) Electric field is in the direction in which the potential decreases the steepest.

ii) Its magnitude is given by the change in the magnitude of potential per unit displacement normal to the

equi potential surface.

Potential energy of a system of charges

Consider two charges q1 and q2 separated by a distance r12. Potential at the position of q2 due to q1 is

12

1

0

1r

q

4

1V

πε= . From the definition of potential, work done in bringing q2 from infinity to the point distant

r12 is q2 times the potential at 12rr

due to q1,

∴ work done on 12

21

0

2r

qq

4

1q

πε=

This work done is stored as potential energy of a system of two charges.

12

21

0 r

qq

4

1U

πε=

If q1q2 > 0, PE is positive. It will be negative q

For a system of three charges.

Total PE

++

πε=

13

3

23

32

12

21

0 r

q

r

qq

r

qq

4

1U

The potential energy is characteristic of the present state of configuration, and not the way

achieved.

4.What is the relation between PE and force?

If the PE is +ve, the force between them will be repulsion. If the PE is

be attractive. If q1 and q2 are both +ve or both

If q1or q2 is +ve and the other –ve the PE will be

F=

Electric field is equal to negative gradient of potential

E=

5. Potential energy of a single charge.

If V is the potential at a point P, which is the work done

upto that point, then work done in bringing a charge q will be qV. This is stored as PE

( )rqVW = or PE = ( )rqV

6. Potential energy of a system of two charges in an external

q1 and q2 are two charges separated by a distance r

an external field. Their total potential energy isis given by

7. Potential energy of a dipole in an external field.

Consider a dipole of dipole moment

torque experienced at this position

angle dθ, is ∆W= pE sinθ dθ.

∴ Total work done in rotating the dipole from an angle

26

0, PE is positive. It will be negative q1q2 < 0

13

13

r

q

The potential energy is characteristic of the present state of configuration, and not the way

.What is the relation between PE and force?

If the PE is +ve, the force between them will be repulsion. If the PE is –ve the force between them will

are both +ve or both –ve then PE will be +ve and the force will be repulsion.

ve the PE will be –ve and the force will be attraction.

F= � ��

��

Electric field is equal to negative gradient of potential

E= � ��

��

Potential energy of a single charge.

If V is the potential at a point P, which is the work done in bringing a unit positive charge from infinity


Potential energy of a system of two charges in an external field.

are two charges separated by a distance r12 residing in a potential V(r1) and V(r

an external field. Their total potential energy isis given by 2211 ()( rVqrVqPE +=

Potential energy of a dipole in an external field.

Consider a dipole of dipole moment a2qprr

= placed at an angle θ in a uniform electric field

torque experienced at this position Eprrr

×=τ . θ=τ sinPE . Work done ∆W in rotating through a small

Total work done in rotating the dipole from an angle θ0 to θ1 is

The potential energy is characteristic of the present state of configuration, and not the way state is

ve the force between them will

ve then PE will be +ve and the force will be repulsion.

ve and the force will be attraction.

in bringing a unit positive charge from infinity


) and V(r2) respectively in

120

212

4)

r

qq

πε+

in a uniform electric field Er

. The

W in rotating through a small

∫θ

θθ−θ=θθ=

1

0

cos(cospEdsinpEW 10

Let us take the zero potential energy as

pcospEcos2

cospEUr

−=θ−=

θ−

π=

Potential energy of the dipole is also given by

where V1 and V2 are the potential at the point where q and

Epa

qpEU

4.

24cos

0

2

πεθ −−=−−=

rr

Since a

q

24 0

2

πε is a constant that term is insignificance for potential energy.

8. Electrostatics of conductors.

i. Inside a conductor, electrostatic field is zero

If there is no current in the conductor electric field is zero. If there is an electric field there will be flow

of charges when the free charges distribute themselves , then electric field becomes zero.

ii. At the surface of a charged conductor, elec

point.

If Er

were not normal to the surface it would have some non

this component free charges will move. Hence field will be normal.

iii. The interior of a conductor can have no excess charge in the static situation

When a conductor is charged excess charge can reside only on the surface. This follows from Gauss law.

Since electric field inside is zero net charge is zero.

iv. Electrostatic potential is constant through out the volume of the conductor and has the same value

(as inside) on its surface

Electric field intensity inside the conductor is zero.

0dv,0dx

dv,0Eif,

dx

dvE ===

−= i.e potential , V= constant.

v. Electric field at the surface of a charged conductor

Where σ is the surface charge density and

direction. Flux contribution from the curved part of the Gaussian surface is zero.

27

)1

This work done is stored as PE.

Let us take the zero potential energy as θ0=2

π

E.prr

Potential energy of the dipole is also given by [ ]a

qVVqU

24 0

2

21 πε−−=

are the potential at the point where q and −q resides.

a

q

24 0

2

πε

is a constant that term is insignificance for potential energy.

Inside a conductor, electrostatic field is zero



At the surface of a charged conductor, electrostatic filed must be normal to the surface at every

were not normal to the surface it would have some non- zero component along the surface. Due to

this component free charges will move. Hence field will be normal.

The interior of a conductor can have no excess charge in the static situation


Since electric field inside is zero net charge is zero.

ectrostatic potential is constant through out the volume of the conductor and has the same value

Electric field intensity inside the conductor is zero.

i.e potential , V= constant.

Electric field at the surface of a charged conductor nE0

)r

εσ

=

is the surface charge density and n)

is a unit vector normal to the surface in the outward

Flux contribution from the curved part of the Gaussian surface is zero.



trostatic filed must be normal to the surface at every

zero component along the surface. Due to

.


ectrostatic potential is constant through out the volume of the conductor and has the same value

is a unit vector normal to the surface in the outward

Flux contribution from the curved part of the Gaussian surface is zero.

On the flat face 00

E,dsq

Edsεσ

=ε

=

vi. Electrostatic shielding

Since all charges resides only on the outer surface of the conductor electric field inside the conductor is

zero. Whatever be the charge and field configuration outside, any cavity in a conductor remains shielded

from outside electrical influence. The

electrostatic shielding.

9. Dielectric and Polarisation:

Polar and non polar molecules:-

In certain molecules centes of positive charge and negative charge coincide. As a result net charge of

molecules becomes zero. Such molecules are said to have symmetric structure. its net dipole moment is

zero. This type of symmetrical molecules are known as non polar molecules. ex. O

There are molecules in which the centers of

a molecule will have net dipole moment. They are known as polar molecules. H

Conductors placed in an external electric field.

When a conductor is placed in an external electric field inside the conductor there will be induced field

(Ein) which is equal and opposite to the external field (E

Dielectric in an external electric field

Dielectrics are non- conducting substances having no free charge carriers. When it is placed in an external

electric field (E0) it induces a field (E

inside the dielectric is non zero.

Non- polar molecule in an external field

In an external electric field, the positive and negative charges of a non

the opposite directions. So that it develops a net dipole moment. The dielectric is said to be polarised by

the external field.

Polarisation( Pr) and Susceptibility(

When a polar or non- polar, dielectric develops a net dipole moment in the presence of an external field,

the dipole moment per unit volume is called polarisation

For linear isotropic dielectrics Pr

ψ=

the electric susceptibility of the dielectric medium.

Unit of Pr

is 23 m

C

m

Cm=

28

0εσ

=



from outside electrical influence. The field inside the cavity is always zero. This is known as

In certain molecules centes of positive charge and negative charge coincide. As a result net charge of


zero. This type of symmetrical molecules are known as non polar molecules. ex. O

There are molecules in which the centers of positive charges and negative charges do not coincide. Such

a molecule will have net dipole moment. They are known as polar molecules. H

Conductors placed in an external electric field.

external electric field inside the conductor there will be induced field

) which is equal and opposite to the external field (E0) so that total field inside the conductor is zero.

Dielectric in an external electric field

nducting substances having no free charge carriers. When it is placed in an external

) it induces a field (Ein) in the opposite direction. Magnitude of E

polar molecule in an external field

In an external electric field, the positive and negative charges of a non- polar molecule are displaced in

So that it develops a net dipole moment. The dielectric is said to be polarised by

) and Susceptibility( ψψψψe)

polar, dielectric develops a net dipole moment in the presence of an external field,

the dipole moment per unit volume is called polarisation Pr

.

Ee

rψ . ψe is a constant characteristics of the dielectric and is known as

the electric susceptibility of the dielectric medium.



field inside the cavity is always zero. This is known as

In certain molecules centes of positive charge and negative charge coincide. As a result net charge of the


zero. This type of symmetrical molecules are known as non polar molecules. ex. O2, H2, N2, CO2 etc.

positive charges and negative charges do not coincide. Such

a molecule will have net dipole moment. They are known as polar molecules. H2O, HCl, CO etc.

external electric field inside the conductor there will be induced field

) so that total field inside the conductor is zero.

nducting substances having no free charge carriers. When it is placed in an external

) in the opposite direction. Magnitude of Ein < E0 so that net field

polar molecule are displaced in

So that it develops a net dipole moment. The dielectric is said to be polarised by

polar, dielectric develops a net dipole moment in the presence of an external field,

is a constant characteristics of the dielectric and is known as

[In linear isotropic dielectrics induced dipole moment is in the direction of the field and proportional to

field strength] σP is the induced surface charge density.

Polarisation tv

qP =

At

qtP =

This polarization is also defined as the induced surface charge per unit area or surface polarization charge

density.

10. Capacity of Capacitance

If charge(Q) on the capacitor is increased its potential (V) increase

ie Q αV or Q= CV or V

QC = where C is a constant known as capacity or capacitance of the conductor.

It is the ability to store charge

faradvolt

Coulomb

V

QC ===

A conductor has a capacity of one farad, if a charge of one coulomb raises its potential by one volt.

11.Principle of a capacitor

Consider A charged metal plate A as in figure (i) a charge Q and potential V. Its capacity is given by

When an uncharged identical metal plate B is brought near A polarization of charge takes place near B as

shown in figure because of this potential of A decreases

charge flows to earth hence potential of A is further lowered, increasing the capacity. This is the principle

of capacitor. A capacitor consists of two conductors, one of them is charged and the other usually

connected. This arrangement is used to increase the capacity of a conductor.

(A single conductor can be used as a capacitor, by assuming other end at infinity)

29


is the induced surface charge density.

At

qt, t is thickness of the slab

A/qP =


If charge(Q) on the capacitor is increased its potential (V) increase

where C is a constant known as capacity or capacitance of the conductor.

A conductor has a capacity of one farad, if a charge of one coulomb raises its potential by one volt.

------

------

as in figure (i) a charge Q and potential V. Its capacity is given by


shown in figure because of this potential of A decreases by increasing the capacity. If we earth B, +ve


of capacitor. A capacitor consists of two conductors, one of them is charged and the other usually

connected. This arrangement is used to increase the capacity of a conductor.




where C is a constant known as capacity or capacitance of the conductor.

A conductor has a capacity of one farad, if a charge of one coulomb raises its potential by one volt. -

as in figure (i) a charge Q and potential V. Its capacity is given by V

Q.


by increasing the capacity. If we earth B, +ve


of capacitor. A capacitor consists of two conductors, one of them is charged and the other usually earth


30

Dielectric Strength : The maximum electric field that a dielectric medium can withstand without break

down (of its insulating property) is called the dielectric strength dielectric strength of air is about 16 Vm103 −× .

12.Parallel plate capacitor

+ + + + + + + + +

A

Region III

Ed

Plate II

Plate I

A parallel plate capacitor consists of two large parallel conducting plates of area A separated by a

distance d (d2 << A). Assume that medium between the plates is vacuum.

Electric field in the outer region I is 022 00

=εσ

−εσ

Electric field in the outer region II is 022 00

=εσ

−εσ

Electric field in the inner region III is 000 22 εσ

=εσ

+εσ

A

QE

00 ε=

εσ

= [since σ =A

Q]

For uniform electric field d

VE = , where V is the potential difference between the plates

−=

ld

dvE

∴ V= Ed=A

Qd

0ε ∴

d

A

V

QC 0ε==

d

AC 0ε=

Capacitance depends only on geometry of the system,

Find the area of plates required to produce 1F capacitance having a separation 1cm is

39

12

2

0

m101085.8

101CdA =

×

×=

ε=

−

−

which is a plate about 30 km in length and breadth. This shows that 1F is

too big a unit in practical purposes.

13.Effect of dielectric on capacitance

When there is vacuum between the plates dEV,E 00

0

0 =εσ

=

When a dielectric slab is completely inserted between the plates it gets polarized pσ and pσ− are induced

surface charge densities ( )

0

p

0

p dEdV,E

ε

σ−σ==

ε

σ−σ=

( )pσ−σ is proportional to σ , ….∴K

p

σ=σ−σ where K >1

∴KA

Qd

K

dV

00 ε=

εσ

=

31

d

KA

V

QC 0ε==

d

KAC 0ε=

k0ε is called the permittivity of the medium and denoted by ε . 0ε=ε K for vacuum K=1,

0ε=ε is called permittivity of the vacuum 0

Kεε

= is called dielectric constant. It is dimensionless

d

KAC,

d

AC 00

0

ε=

ε=

∴0C

CK = .

Hence dielectric constant is the factor by which the capacitance increases from its vacuum value, when

the dielectric is inserted fully between the plates of a capacitor.

Capacity of a capacitor with thin dielectric

A thin dielectric of thickness t, dielectric constant k is inserted as shown in figure.

Capacity is given by

r

0

ttd

AC

ε+−

ε=

t

d

vacuum vacuum

C will be greater then d/AC 00 ε=

If slab is metal ∞=εr , td

AC 0

−

ε=′

C will be greater than C0 = d/A0ε

If slab is metal td

AC, 0

r −

ε=′∞=ε

14. Combination of capacitor

(i) Capacitors in series: Consider two capacitors C1 and C2 connected in series a potential of V voltage

is applied across them. In series connection charge on capacitors are same. Applied voltage V gets

divided as V1 and V2.

2211

21

21 , CVCVQC

Q

C

QVVV ==+=+=

,C

Q

C

Q

C

Q

21

+=21 C

1

C

1

C

1+=

where C is known as effective capacity of the combination 21

21

CC

CCC

+=

for n number of capacitor

n21 V...VVV +++= 21 C

Q

C

Q

C

Q+=

in series connection effective capacity decreases

(2) capacitors in parallel

In parallel connection voltage is same

Total charge stored in the system Q

CV = VCVC 21 + 21 CCC +=

for n capacitors in parallel 1CC +=

15.Energy stored in a capacitor

Consider a capacitor of capacity C. When fully charged potential difference across the capacitor is C and

charge is Q. When the charge in the capacitor is Q

Work done in adding a further charge dQ

∴ total work done in charging from 0 to Q is

This is stored as electrostatic PE

16.Energy density in a parallel plate capacitor

Consider a parallel plate capacitor with capacity

32

nC

Q....++

n21 C

1...

C

1

C

1

C

1+++=

in series connection effective capacity decreases

In parallel connection voltage is same VCQ,VCQ 2211 ==

21 QQ +=

n2 C....C ++


When the charge in the capacitor is Q′ and potential is V′, V′=C

Q′,

charge dQ′ is dW=Vq=V′dQ′ = QdC

Q ′′

total work done in charging from 0 to Q is

∫ =′=Q

0

21

C2

QQd

C

QW

QV

2

1CV

2

1

C2

QW 2

2

=== [since Q=CV]

density in a parallel plate capacitor

Consider a parallel plate capacitor with capacity d

AC 0ε=


[since Q=CV]

33

Energy density ( )

Add

A2

A

CAd2

Q

volume

Energy

0

22

εσ

=== =2

02

1Eε

If there is a dielectric between the plates energy density is given by 2r0 E

2

1U εε= .

17.Potential due to a hollow sphere or solid conducting charged sphere

We have � = ��

�� .

For a hollow sphere of radius “R” carrying a charge “Q”

Electric field intensity outside Eo =

��

�

��

Potential outside is Vo = � �

��[

��

�

��] =

��

�

�

Electric field intensity on the surface Es =

��

�

��

Potential outside is Vs = � �

��[

��

�

��] =

��

�

�

Electric field intensity inside is E i = zero

�� = ��

�� . That is 0 = �

��

��

Then �� = 0 . So �� = �� ℎ�� ℎ � "#$�� .

Then Vi= Vs =

��

�

�

Potential outside is Vs = � �

��[

��

�

��] =

��

�

�

18.What is Van De Graff Generator?

Van de Graff generator is a machine that can build up high voltages of the order of few million volts. The

resulting large electric fields are used to accelerate charge particles (electrons, protons, ions) to high

energies needed for experiments to problem the small scale structure of matter.

19.What is the principle of van De Graff generator?

Consider two concentric sphere shown in figure insulated from each other.

Potential on the surface of sphere of radius R is ( )

+πε

=R

q

R

Q

4

1V

0

R

Potential on the surface of a sphere of radius r is ( )

+πε

=R

Q

r

q

4

1V

0

r

∴Potential difference ( ) ( ) =−4

VV Rr

ie inner sphere is greater potential than outer sphere. This means that if we connect smaller and larger

sphere by a wire, the charge q will flow to the outer sphere.

What ever charge is given to smaller sphere it

charges on the outer sphere until we reach breakdown field of air 93

Van de Graff generator

20.Working of Van de Graff generator.

A Schematic diagram of Van de Graff

supported at height several metres above the ground on an insulating column. A long narrow endless belt

insulating material line rubber or silk is wound around two pulleys

centre of the shell. This belt is kept continuously moving by a motor driving the lower pulley. It

continuously carriers positive charge, sprayed on to it by a brush at ground level, to the top. These it

transistors its positive charge to another conducting brush connected to the larger shell positive charge

spreads out uniformly on the outer surface. In this way voltage differences as much as 6 or 8 million volts

with respect to ground can be built up.

34

−πε R

1

r

1

4

q

0

is +ve.


sphere by a wire, the charge q will flow to the outer sphere.

What ever charge is given to smaller sphere it goes to outer sphere. In this way we can keep piling up

charges on the outer sphere until we reach breakdown field of air 93×106V/m)

.Working of Van de Graff generator.

Graff generator is shown in figure. A large spherical conducting shell is


insulating material line rubber or silk is wound around two pulleys – one at ground



rge to another conducting brush connected to the larger shell positive charge


with respect to ground can be built up.


goes to outer sphere. In this way we can keep piling up

V/m). This is the principle of

generator is shown in figure. A large spherical conducting shell is


one at ground level and the other at



rge to another conducting brush connected to the larger shell positive charge


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