Electrostatics

Class XII/PHYSICS

------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ SRHSS/2010/ Electrostatics of 10 Prepared by: Sandhya.K

ELECTROSTATICS Frictional electricity is the electricity developed on objects, when they are rubbed with each other. The electricity so developed cannot move from one part of the object to other part. Hence, frictional electricity is also called static electricity. Electrostatics is the branch of electronics which deals with the charges and forces resulting from electrons and protons at rest, or static. Charge is a scalar quantity. There are two kinds of charges- positive charge and negative charge Charge originates at the fundamental particle level. For the atomic particles forming stable matter have Proton = +1 unit of charge Electron = -1 unit of charge Neutron = 0 charge A positive charge of an object means that the object has lost electrons. A negative charge of an object means that the object has gained electrons. Properties of electric charge

• Charge can neither be created nor be destroyed. The total charge of an isolated system remains

constant. This is law of conservation of charge.

• Quantization of charge: All existing charges are integral multiples of the fundamental charge, e (1.6 x

10-19 C). The charge on an body is ne± .

• The electric charge is additive: The total charge on an object is the algebraic sum of all the charges

located at different points of the object.

• Like charges repel and unlike charges attract.

For the study of electrostatics we divide materials into two classes: Conductors which allow charge to move freely through them (metals and carbon in which the outermost atomic electrons are “free” to move through the crystalline atomic lattice of the material) Insulators in which charge does not move but stays where it is placed (glass, quartz, mica, rubber, plastics in which the atomic electrons are tightly bound to their nuclei and are not free to move) COULOMB’S LAW The magnitude of the force of attraction or repulsion between two electric charges at rest was studied by Charles Coulomb. He formulated a law, known as "COULOMB'S LAW". Statement According to Coulomb's law: “The electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of charges and inversely proportional to the square of distance between them.” MATHEMATICAL REPRESENTATION OF COULOMB'S LAW Consider two point charges q1 and q2 placed at a distance of r from each other. Let the electrostatic force between them is F.

According to the law:

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Combining above statements:

OR

---------------------(1) Where k is the constant of proportionality which depends on the medium and the units used for charges.

Value of K is equal to04

1εΠ if the medium is air or vacuum, where εεεεo is permittivity of free space .Its

volume is 8.854 x 10-12 c2/Nm2.

22922912

0

1091098755.810854.814.34

141 −−

−×=×=

×××= CNmCNm

πε

Thus in S.I. system numerical value of K is 9 x 109 Nm2c-2. Putting the value of K = 1/4πεπεπεπε0 in equation (i)

)2(−−−−− The force acts along the line joining the charges. FORCE IN THE PRESENCE OF DIELECTRIC MEDIUM If the space between the charges is filled with a non conducting medium or an insulator called "dielectric", it is found that the dielectric reduces the electrostatic force as compared to free space by a factor called

DIELECTRIC CONSTANT. It is denoted by εr. This factor is also known as RELATIVE PERMITTIVITY. It has different values for different dielectric materials. In the presence of a dielectric between two charges the Coulomb's law is expressed as:

)3(221

41

−−−−−−−−⋅=r

qq

medFπε

, where ε is called absolute permittivity of the

medium. Dividing equation (2) by equation (3),

)4(0

221

41

221

041

−−−−−−−−−−=

⋅

⋅

=εε

πε

πε

r

qqr

qq

medFvacF

0εε

is denoted rε , called relative permittivity of the medium with respect to the medium. It is also denoted by

K, called dielectric constant of the medium. Therefore, equation (4) gives,

)5(0

−−−−==medFvacF

r εε

ε

Thus, relative permittivity or dielectric constant of a medium may be defined as the ratio of force between two charges placed at a certain distance apart in air to the force between the same two charges placed the same distance apart in that medium. VECTOR FORM OF COULOMB'S LAW

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Consider two point charges q1 and q2 separated by a distance r. If q1 q2 > 0 i.e. if both q1 and q2 are +ve

or both q1 and q2 are negative then the charges repel each other otherwise they attract each other.

So the forces exerted by charges on each other are equal in magnitude opposite in direction

PRINCIPLE OF SUPERPOSITION The principle of superposition gives a method to find force on a charge, when a group of charges are interacting. It states that- “when a number of charges are interacting, the total force on a given charge is the vector sum of the individual forces exerted on the given charge by all the other charges.”

Consider n point charges q1, q2, q3, ……qn are distributed in space. The charges are interacting with each other.

Let us find the force on q1due to all other charges. If the charges q2, q3, ……qn exert forces 12Fr

, 13Fr

, …. nF1r

on the charge q1 (fig), then according to principle of superposition, the total force on charge q1 is given by

)6(....... 113121 −−−−−−−−−−−+++= nFFFFrrrr

If the distance between the charges q1 and q2 is denoted as 12r ; and 12r̂ is unit vector from charge q2 to q1,

then 21212

21

012 ˆ

41

rrqq

F ⋅=πε

r

Similarly, the force on charge q1 due to other charges is given by

31213

21

013 ˆ

41

rr

qqF ⋅=

πε

r

…………………………………………… ……………………………………………..

121

21

01 ˆ

41

n

n

n rr

qqF ⋅=

πε

r

Hence, substituting for 12Fr

, 13Fr

, …. nF1r

in equation (6), the total force on the charge q1 due to all other

charges is given by-

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)7(ˆ........ˆˆ41

....... 121

1312

13

31212

12

21

0113121 −−−−−−−−−

+++=+++= n

n

nn r

rqq

rrqq

rrqq

FFFFπε

rrrr

The same procedure can be adopted for finding the force on any other charge due to remaining charges of the group of n charges.

Question and Answers

1. What does q1+q2=0 signify in electrostatics?

A. The charges q1 and q2 are equal and opposite. 2. Electrostatic force between two charges is called central force. Why? A. The force between two charges always acts along the line joining the two charges. For this reason it is

called central force. 3. In Coulomb’s law, on what factors the value of electrostatic force constant depends? A. On the system of units adopted and the nature of medium, in which two charges are placed. 4. If the distance between two equal point charges is doubled and their individual charges are also

doubled, what would happen to the force between them? A. Originally, the force between the two charges,

24

1

0 r

qqF

×⋅=

πε

When the individual charges and the distance between the charges are doubled, the force is given by-

( )22

2241

0 r

qqF

×⋅=

πε=

24

441

0 r

qqF

×⋅=

πε=

241

0 r

qqF

×⋅=

πε=F

That is force will remain the same.

5. A comb run through one’s dry attracts small bits of paper. Why? What happens, if the hair is wet or if it is a rainy day?

A. When a comb is run through dry hair, it gets charged. As such, it starts attracting small bits of paper. If the hair is wet or it is rainy day, then in absence of friction, the comb will not get charged, when run through hair. In absence of charge on the comb, the bits of paper will not be attracted.

6. Is Coulomb’s law in electrostatics applicable in all situations? A. No, Coulomb’s law in electrostatics does not hold in all the situations. It is applicable only in the

following situations- a) The electric charges must be stationary. b) The electric charges must be points in size.

ELECTROSTATIC FIELD (Er

) Space or region surrounding an electric charge or a charged body within which another charge experiences some electrostatic force of attraction or repulsion when placed at a point is called Electric Field.

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ELECTRIC FIELD INTENSITY Electric intensity is the strength of electric field at a point. Electric field intensity at a point is defined as the force experienced per unit positive charge at a point placed in the electric field. Or It may also be also defined as the electrostatic force per unit charge which the field exerts at a point. Mathematically,

Electric field strength is a vector quantity directed away from a positive charge and towards the negative charge. SI unit of electric field is newton/coulomb (NC−1) or volt metre (Vm−1). The dimensional formula for E is MLT−3A−1.

The electric field due to a source point charge q at a distance r from it is given by-

ELECTRIC FIELD DUE TO A POINT CHARGE / ELECTRIC INTENSITY DUE TO A POINT CHARGE Consider a point charge q called SOURCE CHARGE placed at a point ‘O’ in space. To find its intensity at a point ‘p’ at a distance ‘r’ from the point charge we place a test charge 'q'.

The electric field intensity will be:

qF

E′

= ----(1)

According to coulomb's law the electrostatic force between them is given by:

24

1

0 r

qqF

′=

πε-----------(2)

Substituting equation (2) in eqn(1)

We get: qF

E′

= or Fq

E ×′

=1

Putting the value of 'F' we get:

200 4

124

11rq

r

qqq

E ×=′

×′

=πεπε

This shows that the electric intensity due to a point charge is directly proportional to the magnitude of charge q and inversely proportional to the square of distance. In vector form,

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rr

qr

r

qE

rr

3041ˆ

2041

⋅=⋅=πεπε

ELECTROSTATIC LINES OF FORCE: PROPERTIES A line of force is an imaginary path straight or curved such that the tangent to it at any point gives the direction of electrostatic field at that point. Or A field line is an imaginary line along which a unit positive charge would move when set free. The lines of force are drawn such that the number of lines per unit area of cross-section, (area held normally to the field lines) is

proportional to magnitude of Er

.

PROPERTIES OF FIELD LINES a) Field lines always come out of positive charge and enter the negative charge.

b) Field lines never cross each other. c) Field lines never form closed loops. d) Field lines are always directed from higher potential to lower potential. e) Field lines never exist inside a conductor. f) Field lines always enter or leave a conducting surface at right angles. g) Tangent to field line at a point gives the direction of field at that point. h) Electric lines of force contract longitudinally. Field Lines in Some Cases

(a) Positive point charge

Field lines have spherical symmetry

(b) Negative point charge

Field lines have spherical symmetry

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(c) Two similar charges of equal magnitude

(e) Two dissimilar charges of equal magnitudes

(d) Two similar charges of unequal magnitudes

ELECTRIC DIPOLE A system of two equal and opposite charges separated by a small distance is called an electric dipole.

The molecules of water, ammonia, etc behave as electric dipoles. ELECTRIC DIPOLE MOMENT It is defined as the product of either of charge and the distance between the two charges. )2( aqp

rr=

The dipole moment p

r is a vector quantity. It is directed from negative charge to positive charge.

Unit of dipole moment (p) = coulomb × metre = C.m

Dimensions of p = M0L1T1A

1

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ELECTRIC FIELD AT AN AXIAL POINT OF ELECTRIC DIPOLE

Consider an electric dipole consisting of charges +q and –q, separated by a distance 2a and placed in free space. Let P be a point on the line joining the two charges (axial line) at a distance r from the centre O of the dipole.

The electric field Er

at point P due to the dipole will be the resultant of the electric field AEr

(due to charge –q at

point A) and BEr

(due to charge –q at point B) i.e.

Er

= AEr

+ BEr

Now, ( )20

20 4

141

arq

APq

EA +⋅=⋅=

πεπε

r (along PA)

And ( )20

20 4

141

arq

BPq

EB −⋅=⋅=

πεπε

r (along PX)

Obviously, BEr

is greater than AEr

. Since AEr

and BEr

act along the same line but in opposite direction, the

magnitude of the electric field at point P is given by,

E= Er

= AB EErr

− (along PX)

Or ( ) ( )

( ) ( )( ) ( )222

0222

22

0

20

20

)4(41

41

41

41

ar

raq

ar

ararqE

arq

arq

E

−⋅=

−

−−+⋅=

+⋅−

−⋅=

πεπε

πεπε

Now, q(2a)=p, the magnitude of the electric dipole moment of the dipole.

( )2220

241

ar

prE

−⋅=∴

πε (along PX)

In vector notation,

( )2220

241

ar

rpE

−⋅=∴

rr

πε

When dipole length is very small, r>>>a, then a2 can be neglected as compared to r

2. Therefore for an electric

dipole of very small length,

30

40

2412

41

rp

rpr

E ⋅=⋅=πεπε (along PX)

ELECTRIC FIELD AT AN EQUATORIAL POINT

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Consider an electric dipole consisting of charges +q and –q, separated by a distance 2a and placed in free space. Let P be a point on the equatorial line of the dipole at a distance r from the centre of the dipole. (as shown in figure).

Let AEr

and BEr

be the electric fields at the point P due to charge –q at A and charge +q at point B. Then,

resultant electric field at point P is given by

Er

= AEr

+ BEr

Now, 220

20 4

141

arq

APq

EA +⋅=⋅=

πεπε

r (along PA)

And 220

20 4

141

arq

BPq

EB +⋅=⋅=

πεπε

r (along BP)

AEr

and BEr

have same magnitude. To calculate the resultant

electric field due to the dipole at point P, we can use the

triangle law of vector addition. In ∆PAB, PBAPrr

, and ABr

represent AEr

, BEr

and Er

respectively. Therefore by the law

of addition of vectors,

BP

E

PA

E

BA

E BA

rrr

==

Or, PABA

EE A ×=rr

=

( )21

2222

0

241

ar

aarq

+×

+⋅

πε

Or,

( ) 23

220

)2(41

ar

aqE

+⋅=

πε

Now, q(2a)=p, the magnitude of the electric dipole moment of the dipole.

( ) 232204

1

ar

pE

+⋅=∴

πε (along PX’)

In vector notation, ( ) 2322

041

ar

pE

+⋅=

rr

πε

When dipole is of very small length, that is, a<<<r, a2 can be neglected as compared to r

2. Therefore for an

electric dipole of very small length,

304

1rp

E ⋅=∴πε

ELECTRIC DIPOLE IN UNIFORM ELECTRIC FIELD

Consider an electric dipole consisting of charges –q and +q and of length 2a placed in an uniform electric field

Er

, making an angle θ with the direction of the field as shown in figure above.

Force on the charge –q = -qEr

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And force on charge +q = +qEr

Thus the electric dipole is under the action of two equal and unlike parallel forces, and gives rise to a torque on the dipole. The magnitude of the torque is given by, τ = either force x perpendicular distance between them = qE x AN = qE x (2a sinθ) = q (2a) E sinθ Or τ = pE sinθ , where q (2a) =p, is the electric dipole moment of the dipole. The torque tends to align the dipole along the field direction. Since the electric dipole moment vector p

ris a vector from the charge –q to +q, the equation may be expressed

as,

Eprrr

×=τ

• When the dipole is placed in a uniform electric field, it experiences only a torque. Net force on the dipole is zero.

• Torque on the dipole becomes zero, when it aligns itself parallel to the electric field. i.e., when θ=0, τ = p E sin0= 0 • Torque on the dipole is maximum, when the dipole is placed at right angles to the direction of the

electric field. i.e., when θ=90°, τ = p E sin90°= pE Questions and Answers

1) Electric field intensity within a conductor is always zero. Why? A. The electric lines of force cannot pass through a conductor. As a result, there can be no electric field inside a conductor. Hence, the electric field intensity inside a conductor is zero. 2) Are the filed lines a reality? A. The electric field lines are purely geometrical constructions, which are used to represent the electric field graphically. The electric field lines are imaginary but the electric field they represent is real.

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