ELECTRO STATICS

ELECTIC CHARGES AND FIELD

Electrostatics is a branch of physics which deals with the electric charges which are at rest.
It is also known as static electricity.
Note : The devices which works under electrostatic are
1. Cathode ray tube.
2. Vandegraff generator.
3. Capacitors (used in radio, T.V. radars)
4. Pollution is checked by the method of electrostatics.
Electric Charge : It is a fundamental property of matter which repels its own kind and attracts its
opposite kind.
Note:
1. The SI unit of charge is coulomb „C’.
2. The dimensional formula of charge is [AT]
3. The apparatus used to find the presence and nature of electric charge is Electroscope.
[The most commonly used is gold leaf electroscope]
4. A nylon garment often crackles when it is taken off
5. A thin stream of water from a tap can easily be deflected by electrified rod.
6. During landing the tyres of an air craft get electrified .Therefore special material is used to
manufacture the tyres.
7. Electric charges associated with electrons are called conventional –ve charge. Electric charges
which are associated with protons are called conventional +ve charge.
8. Neutrons are electrically neutral.
9. The gravitational force associated with charges is neglected because the electro static force between
the charges is very high compare to it.
10. The elementary charge of electron or proton is 1.6 × 10-19 C.
11. The no of electrons present in 1 C of charge is ne = q
ne = 1C
n = 1/e
n = 1/ 1.6 × 10-19 C
n = 6.25 ×10 18
12. In an atom always equal no of protons and equal no of electrons are present in its ground state.
Therefore atom is electrical neutral.

Conductors: These are the substance which allows the electric charges to flow through them easily.
Ex : All metals , human body, earth, humid air , Impure water---etc.
Conductors have more no of free electrons.
1
Insulators: The substances which does not allow charges to flow through them easily are called insulators.
Ex : Dry wood, Rubber, Paper, plastics pure water ebonite, mica glass, porcelin, nylon.
Insulators have bound electrons.

Distribution of charges:
1. For a conductor of uniform surface the distribution of charge is also uniform.
2. For a conductor of non uniform surface the distribution of charge is also non uniform
3. The distribution of charge is maximum, where the curvature is maximum
4. The distribution of charge is minimum, where the curvature is minimum.
5. The distribution of charge is maximum, where the radius of curvature is minimum.
6. The distribution of charge is minimum, where the radius of curvature is maximum.
7. The distribution of charge is maximumat a sharp edge or pointed end.

Methods of charging a body : The process of adding or removing the charges from a body is known as
charging or electrification. A body can be charged by the following methods.
 Friction method.
 Conduction method.
 Induction method.

Friction method: When two suitable bodies are rubbed together the mutual transaction of charges takes place.
One body gains charges and the other body looses the charges. The body which gains charges will become
negatively charged and the body which looses charges will become positively charged. This method of
charging a body is called friction.
In this method the law of conservation of charges holds good.
Positively charged materials Negatively charged material
1. Glass rod silk
2. Fur rubber
3. Dry hair Plastic comb
4. wool plastic
Charging by Conduction : When a uncharged body is bought in contact with a charged body, the uncharged
body gains charges. This method of charging a body is called conduction.
In this method both charged body and uncharged body will have same kind of charges
Charging by induction : The process of charging a body by bringing a uncharged body closed to a charged
body is called induction method . In this method both the bodies will have opposite kind of charges.

Consider two spheres A and B supported on insulating stand in contact.

2
Bring a positively charged rod near one of the sphere A, the free electron of A are attracted to the rod. This
leaves an excess +ve charge on rarer surface of B. both kind of charges are bound to the spheres and cannot
escape. In a short time equilibrium is reached under the action of force of attraction of the rod and the force of
repulsion of due to the accumulated charges. This process is called induction method.
Separate the two spheres by a small distance while holding the glass rod near the sphere A. the spheres found
to be oppositely charged and attract each other.

Remove the rod the opposite charges of A and B come close to each other and attract each other

Separate the sphere apart, the charges on them get uniformly distributed over them. In this process the spheres
will be equally and oppositely charged. this is charging by induction.

Basic Properties of electric charges:

1. Charge is additive in nature ( Charges are added and subtracted )
2. In a closed system electric charges are always conserved.
3. Charges are quantized. It exists in the form of discrete packets.The charges on a body is equal to the
integral multiple of basic unit of charge
𝒒 = ±𝒏𝒆
Where, n integer, e elementary charge.
4. Charges are always associated with mass.
5. Like charges repel and unlike charges attract.
6. Charge is a scalar quantity.
7. Charges always reside on the surface of a conductor..
8. Charge is independent of the velocity of the body
9. Accelerated charge radiates energy in the form of electromagnetic waves.
Note : 1. If electrons are removed from a body then the body becomes electrically positive.
2. If some electrons are added to the body then the body becomes electrically negative.
3
Quantization of charges: existence of charges is discrete packets rather than in continuous, electrical charge
is always integral multiple of elementary charge this is known as quantization of charge.
Discharging action at a point :
When a pointed end conductor is charged continuously more no of charges are accumulated at the sharp edge
because of this mutual repulsion of charges takes place and some of the charges are leaked out to the
surrounding system. This is called discharging action at a point.
Note :
1. Discharging action at a point forms the principle of construction of lightening arrestors, Vandegraff
generators
2. Insulators are also called as dielectric.
3. Charges on clouds are due to the motion of water droplet in the atmosphere.
4. When a +ve charged body is connected to the earth electrons flow from the earth to the body and
neutralize the body.
5. When a –ve charged body is connected to the earth electrons flow from the body to the earth and
neutralize the body.
Earthing: when a charged body is brought in contact with the earth, all the excess charges on the body
disappears by causing momentary current to pass to the ground through the body. This process is called
grounding.
Point Charge : Point charge is a charged body whose size is negligible, when compare to their distance of
separation.
Coulomb’s law : (inverse square law of electrostatic )
The coulombs law states that “ The force of attraction or repulsion between any two isolated point charges is
directly proportional to product of their magnitudes and inversely proportional to square of the distance
between them”

q1 q2
r
Consider two charged bodies A and B separated by a distance „r‟. Let 𝑞1 and 𝑞2 be the magnitude of charges
on A and B and F is the force between them.
From coulomb‟s law
𝐹  𝑞1 𝑞2 -----1
1
𝐹 -----2
𝑟2

Compare 1 and 2
𝑞1 𝑞2
𝐹
𝑟2
where, K constant of proportionality.

4
 1
For air 𝐾= = 9 × 109 𝑁𝑚2 𝐶 −2
4𝜋𝜀 0

𝟏 𝒒𝟏 𝒒𝟐
𝑭=
𝟒𝝅𝜺𝟎 𝒓𝟐
Note :1. o is the obsolete permittivity of free space
o = 8.354 X 10-12 C2 N-1 m-2 [Fm-1]
2. The dimensional formula for o is [ M-1 L -3 T 4 A 2 ]

Coulomb’s law in Vector form:

Case 1 : when 𝑞1 𝑞2 > 0

consider two like charges 𝑞1 𝑞2 > 0 , then there will
be a force of repulsion between them.
The coulomb force between q1 and q2 is
𝟏 𝒒𝟏 𝒒𝟐
𝑭𝟐𝟏 = 𝒓
𝟒𝝅𝜺𝟎 𝒓𝟐𝟐𝟏 𝟐𝟏

Where 𝐹21 → 𝑓𝑜𝑟𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑞1 𝑎𝑛𝑎 𝑞2

𝑟21 → 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑞1 𝑎𝑛𝑑 𝑞2
𝑞1 𝑎𝑛𝑑 𝑞2 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑐ℎ𝑎𝑟𝑔𝑒𝑠
Case 2: when 𝑞1 𝑞2 < 0
Consider two like charges q1q2< 0 , then there
will be a force of attraction between them
The force experienced by q1 due to q2 is
𝟏 𝒒𝟏 𝒒𝟐
𝑭𝟏𝟐 = 𝒓
𝟒𝝅𝜺𝟎 𝒓𝟐𝟏𝟐 𝟏𝟐
Note: 1. F12 = - F21

Force between multiple charges:

Consider a system having multiple charges like q 1, q2 …..qn placed

in vacuum, then the force between these charges are formed using
superposition principle.

5
The force on any charge due to number of other charges is the vector sum of all the forces on the
individual charges due to the other charges taken one at a time. The individual forces are unaffected
due to the presence of other charges.
Force on q1 due charge q2
𝟏 𝒒𝟏 𝒒𝟐
𝑭𝟏𝟐 = 𝒓
𝟒𝝅𝜺𝟎 𝒓𝟐𝟏𝟐 𝟏𝟐
Force on q1 due charge q3
𝟏 𝒒𝟏 𝒒𝟐
𝑭𝟏𝟑 = 𝒓
𝟒𝝅𝜺𝟎 𝒓𝟐𝟏𝟑 𝟏𝟑

In general

𝒏
𝒒𝟏 𝒒𝒊
𝑭= 𝒓𝒊
𝟒𝝅𝜺𝟎 𝒓𝟐𝟏𝒊
𝒊=𝟐

Factors on which coulomb force between the point charges depends :

The force b/n two point charges depends upon,
1. The magnitude of charges ( F  q 1 q 2 )
2. Distance b/n the charges ( F  1 / r2 )
3. The medium in which charges are kept ( F  1 / K )
Limitations of coulombs law :
The coulombs law is applicable only for isolated point charges .

Unit Charge :
Unit charge is that charge which when placed in air by a distance of 1m from an equal and similar
charge repel with the force of 9 × 109N .

Define SI unit of charge or Coulomb: A charge is said to be one coulomb which when placed in air or
vacuum at distance of 1m from an identical charge repels with a force 9 X 109 N.
𝟏 𝒒𝟏 𝒒𝟐
𝑭=
𝟒𝝅𝜺𝟎 𝒓𝟐
𝑤ℎ𝑒𝑛 𝑞1 = 1𝐶 𝑎𝑛𝑑 𝑞2 = 1𝐶 , 𝑟 = 1𝑚, 𝑡ℎ𝑒𝑛𝐹 = 𝟗 × 𝟏𝟎𝟗 𝐍

6
Relative permittivity of a medium or dielectric constant ( εr ) :
𝑭𝒂
= 𝜺𝒓
𝑭𝒎
Dielectric constant of a medium is defined as the ratio of force between the two point charges separated
by a distance in air to the force between same two point charges separated by the same distance in a
medium.
Note: For air dielectric constant is 1 for any other media other than air dielectric constant is greater than one
Substance Dielectric constant.
Wax 2
Glass 3 or4
Mica 7
Glycerin 39
Water 80
The Coulomb force between the two isolated point charges separated by a distance in a medium is given by,
𝟏 𝒒𝟏 𝒒𝟐
𝑭=
𝟒𝝅𝜺𝟎 𝑲 𝒓𝟐
ε = Kεo K relative permittivity of the medium.
Electric field is a region where a test charges experiences a force.
1. If the force experienced by the test charge is same at all the points in an electric field then the electric field
is said to be uniform
2. If the force experienced by the test charge is different at different points then the electric field is said to be
non uniform.
3. If a test charge experiences no force in the region then the electric field is said to be zero.
Electric field or electric intensity: (E)
Electric field due to a source charge at a point is defined as the force experiences by a test charge placed
at that point.
Consider a test charge q is placed at a point in an electric field. If „F‟ is the force experienced by the test
𝑭
charge then, the electric intensity at that point is 𝑬 = 𝒒

Note : 1.The SI unit of electric intensity is NC-1

2. Electric intensity is a vector quantity
3. The dimensional formula of electric intensity is [ M1L1 T-3 A-1]
4. The force experienced by a test charge in the electric field is F = E qo . If qo is a unit +ve charge then
force experienced by the charge is numerically equal to the electric field strength or electric intensity.
Expression for electric intensity due to an isolated point charge:(E)
Q q

r P
7
Consider an isolated point charge of magnitude „Q‟ kept in free space. Let „P‟ be a point at a distance of „r‟
from the charge „Q‟. Let a test charge of magnitude „q‟ is placed at the point „P‟. From coulombs law the
force experienced by the test charge is,
𝟏 𝑸𝒒
𝑭= 1
𝟒𝝅𝜺𝟎 𝒓𝟐

from the definition of electric field

𝐹
𝐸= →2
𝑞
From eqn 1 and 2,
𝟏 𝑸𝒒
𝟒𝝅𝜺𝟎 𝒓𝟐
𝑬=
𝐪
𝟏 𝑸
𝑬=
𝟒𝝅𝜺𝟎 𝒓𝟐
Note : 1. If the given charge is +ve then the electric intensity is directed away from the charge.
+q E

2. If the given charge is –ve then the electric intensity at a point is directed towards the charge.
-q E

3. The electric intensity in a medium is given by

𝟏 𝒒
𝑬=
𝟒𝝅𝜺𝟎 εr 𝒓𝟐
4. Electric Intensity due to an isolated point charge is inversely proportional to square of the distance.
5. If number of electric fields are acting at a point due to charges then the net electric field at that point
is given by
E = E1 + E2 + E3 + ---------------
6. If E1 and E2 are the two electric fields due to two charges acting at a point and  is a angle b/n them.
Then the net electric intensity at that point is given by

𝐸 = 𝐸12 + 𝐸22 + 2𝐸1 𝐸2 𝑐𝑜𝑠

The direction of the net electric field is given by

𝑬𝟐 𝐬𝐢𝐧 𝜽
𝜶 = 𝒕𝒂𝒏−𝟏
𝑬𝟏 + 𝑬𝟐 𝐜𝐨𝐬 𝜽
7. If we assume the test charge are very very small positive test charges then
𝑭
𝑬 = 𝐥𝐢𝐦
𝒒𝟎 →𝟎 𝒒𝟎

Note: Electrical field is spherically symmetric at any point at a distance „r‟ from the source charge; the
magnitude of electric field remains the same.

8
Electric field due to system of charges:

Electric field at a point in space due to system of charges is defined to be, the force experienced per unit test
charge placed at that point without disturbing the other source charges.
Consider, a system of charges q1, q2….qn with position vector r1P, r2P……rnp.
Electric field E1 at r due to q1 is

Electric field E2 due to charge q2 is

For nth charge electric field is

𝟏 𝒒𝒏
𝑬𝒏 = 𝒓
𝟒𝝅𝜺𝟎 𝒓𝟐𝒏𝒑 𝒏𝒑
From the superposition principle Electric field at‟ r‟ due to system of changes is,
𝑬 = 𝑬𝟏 + 𝑬𝟐 + … … . + 𝑬𝒏

Electric field line: Electric field lines are curved imaginary lines in the electric field such that the tangent at
any point on the field line gives the direction of the electric field at that point.
Faraday introduced the concept of electric lines of forces.

9
The electric field lines around a +ve charge are directed radially outwards and around a –ve charge the lines
are directed radially inwards.
Electric field lines due to two positive charges

Electric field lines between a positive charge and a negative charge

Properties of Electric Field lines:

1. Electric field lines are diverges at a +ve charge and converges at a –ve charge
2. Electric field lines starts at a +ve charge and ends at –ve charge
3. Two electric field lines never intercept.( nor cross each other )
4. The Electric field lines are always perpendicular to the surface of a charge conductor.
5. Electric field lines exert lateral pressure on each other.
6. A tangent drawn to an Electric field line at any point gives the direction of electric field at that point.
7. Electric field lines do not pass through a conductor.
8. Electric field lines pass through a non conductor or insulator.
9. Electric field lines are parallel in a uniform electric filed.
10. Electric field lines are crowded in a strong electric filed.

Note: 1. The number of electric field line per unit area crossing a surface at right angle to the direction of
the filed is proportional to the electric intensity.
2. Electric field line is a curve drawn in such a way that the tangent at it at each point gives the
direction of net field at that point.

Electric dipole : A pair of equal and opposite charges separated by a small distance is called an electric dipole.

+q -q
2a

10
Electric dipole moment : (P)Electric dipole moment is defined as the product of any one charge and the
distance of separation b/n the charges.

+q -q
2a
the electric dipole moment P = q × 2a
P = 2a × q
Note: 1. Direction of electric dipole moment is from negative charge to positive charge.
2. The SI unit of electric dipole moment is Cm. (coulomb meter)
Electric field at apointdue to an Electric dipole along the axial line:

Consider, a dipole separated by a distance 2a. Let O be the centre of the dipole.
Consider a point P at a distance r from the centre of the dipole along the axial line.
The resultant electric field at P due to two charges is,
𝑬 = 𝑬−𝒒 + 𝑬+𝒒
Electric field at P due to charge –q,
𝟏 𝒒
𝑬−𝒒 =
𝟒𝝅𝜺𝟎 𝑩𝑷𝟐
𝟏 𝒒
𝑬−𝒒 = 𝟐
𝒂𝒍𝒐𝒏𝒈 𝑷𝑨
𝟒𝝅𝜺𝟎 𝒓 + 𝒂
Electric field at P due to charge +q,
𝟏 𝒒
𝑬+𝒒 =
𝟒𝝅𝜺𝟎 𝑨𝑷𝟐
𝟏 𝒒
𝑬+𝒒 = 𝟐
𝒂𝒍𝒐𝒏𝒈 𝑩𝑷
𝟒𝝅𝜺𝟎 𝒓 − 𝒂
The electric field E-q and E+q are acting in opposite direction at P and
𝐸+𝑞 > 𝐸−𝑞
𝑬 = 𝑬+𝒒 + ( − 𝑬−𝒒 )
𝑬 = 𝑬+𝒒 − 𝑬−𝒒

q  1 1 
Eaxial    
4 0   r  a 2  r  a 2 

q  (r  a) 2  (r  a) 2 
Eaxial   
4 0   r  a 2  r  a  2 

11
  
q  r 2  a 2  2ar  r 2  a 2  2ar 

4 0   r 2  a2  
2

 
q 4ar
 ∵ 𝑃 = 2𝑎𝑞
4 0  r 2  a 2 2

2 Pr
Eaxial 
4 0  r 2  a 2 
2

For r  a , neglect a
2 Pr

4 0 r 4

1 2P
Eaxial 
4 0 r 3
Note: 1. Electric field varies as 1/r3 in case of dipole and 1/r2 in case of isolated point charge.
Ex: In the molecule of NH3 H2O HCL the centre of +ve charge and –ve charge separated by a distance.

Electric field at a point on the equatorial plane due to a dipole.

Consider, a dipole separated by a distance 2a.Let a point P on the equatorial plane at a distance „r‟ „from the
centre of the dipole i.e OP is a perpendicular bisector of the length of the dipole.
Electric field at the point P due to charge –q
𝟏 𝒒
𝑬−𝒒 =
𝟒𝝅𝜺𝟎 𝑩𝑷𝟐

Electric field at the point P due to charge +q

𝟏 𝒒
𝑬+𝒒 =
𝟒𝝅𝜺𝟎 𝑨𝑷𝟐
𝟏 𝒒
𝑬+𝒒 =
𝟒𝝅𝜺𝟎 𝒓𝟐 + 𝒂𝟐
1 q
E q  E q  E 
4 0 r  a 2
2

The resultant electric field at P is obtained by resolving E-q and E+q into two components. The component E-q
sin and E+q sin act along equatorial line. They are equal and opposite. They cancel each other. The
component E-qcos and E+qcos act along the same direction PX but opposite to the direction of dipole
moments.

Eeq = − 𝑬−𝒒 𝒄𝒐𝒔𝜽 + 𝑬+𝒒 𝒄𝒐𝒔𝜽

Eeq = − 𝑬−𝒒 + 𝑬+𝒒 𝒄𝒐𝒔𝜽

12
 Eeq  2 E cos 
From diagram
𝒂
𝒄𝒐𝒔𝜽 = 𝟏
𝒓𝟐 + 𝒂𝟐 𝟐

−𝟐𝒒 𝒂
𝑬eq= 𝟏
𝟒𝝅𝜺𝟎 𝒓𝟐 +𝒂𝟐
𝒓𝟐 +𝒂𝟐 𝟐

−𝟐𝒒𝒂
𝑬eq= 𝟑/𝟐 ∵ 𝑃 = 2𝑎𝑞
𝟒𝝅𝜺𝟎 𝒓𝟐 +𝒂𝟐

If r >>a 2, a can be neglected.

−𝑷
∴ 𝑬𝒆𝒒 =
𝟒𝝅𝜺𝟎 𝒓𝟑
Along PX
Note : 1. 𝐸𝑎𝑥𝑖𝑎𝑙 = 2𝐸𝑒𝑞𝑙 That is electric field due to dipole at any point on the axial line is twice the electric
field due to dipole on the equatorial line.
2. Electric field due to a dipole depends not only on the distance „r‟ but also the angle between the
dipole moment.
3. When the dipole size 2a approaches zero the charge q approaches infinity in such a way that the
product P  q  2a is finite. Such a dipole is referred to as a point.
Dielectrics : The substance which do not conduct are called dielectrics. Ex: Paper, plastic, mica. etc.,
Dielectrics are classified into two types they are
 Polar dielectrics: A molecule in which centre of +ve charge does not coincide with the centre of –ve
charge when no external field is applied is called polar dielectric.
Ex: HCL, H2O,NH3, CO2, Alcohol. etc.,
 Non Polar dielectric: A molecule in which the centre of +ve charge coincide with centre of –ve
charge is called non – polar dielectric .
Ex: CO2, CH4, N2O2
Note : Water has polar molecules and transformer oil has non polar molecule. Therefore transformer oil is
better dielectric.
Torque in an electric dipole ()
The product of either of forces and perpendicular distance between the forces is called torque on an electric
dipole.
Torque due to dipole in an uniform electric field
Consider an electric dipole in an uniform electric field „E‟. A force +Eq acts
on the +ve charge and a force of –Eq acts on the –ve charge due to electric
field. The net force produced is  Eq  Eq  0 . These two equal forces acting
on a dipole constitute couple. This couple produces rotating effect called
torque.

13
  = force x perpendicular distance
 = E q 2a sin
= P E sin
Note: consider,  = P E sin
1. When  = 0 i.e., when dipole moment is parallel to the electric field  = 0
2. When  = 1800 i.e., when dipole moment is anti parallel to the electric field  = 0
3. When  = 900  = P E i.e., torque is maximum.
4. The SI unit of torque is Nm.

Torque due to dipole in a Non-uniform electric field:

When a dipole is placed in a non uniform electric field, the dipole experiences both force and torque i.e, force
is non zero. When P is parallel to E the torque acting on the dipole is zero.

The dipole has net force in the direction of increasing field.

When P is anti parallel to E,

The net force on dipole is in the direction of decreasing field.

Continuous change distribution:
Continuous change distribution is considered on a charged body of reasonable size where the charge on a
charged body is so large as compared to the magnitude of a charge on are electron that t quantization of charge
may be ignored.
Linear charge density: ( )
When charges are distributed along a line, the charge distribution is called linear.
Then charge per unit length is called linear charge density
𝒒
𝝀=
𝑳
q Uniform distribution of charge
L Length of the wire.
The SI unit of 𝜆 is Cm-1.
Note: If q charges are distributed uniformly over a circular ring of radius R then,
𝒒
𝝀=
𝟐𝝅𝑹
14
Surface charge density:( σ)
Charge per unit area when the charges are uniformly distributed over an area is called surface charge density.
It is given by,
𝒒
𝝇=
𝑨
Q  charge distributed over a plane surface.
A  Area.
The SI unit is Cm-2.
Note: If charge „q‟ is uniformly distributed over the spherical surface of the conductor of radius R then,
𝒒
𝝇=
𝟒𝝅𝑹𝟐
Note: If „n‟ drops of a liquid carrying a charge „q‟ are combine to form a single drop then the ratio of surface
densities is
𝝇𝒃
= 𝒏𝟏/𝟑
𝝇𝒔
Volume charge density: (ρ)
Charge per unit volume when the charges are uniformly distributed over the volume of the object is called
volume charge density.
𝒒
𝝆=
𝑽
q  charge distributed over the volume.
V  volume.
The SI unit of is Cm-3.
Note: If q is uniformly distributed over the entire volume of the sphere then,
𝒒
𝝆=𝟒
𝝅𝑹𝟑
𝟑

Note : The electric field due to the charge ρV is,

𝟏 𝝆𝑽
𝑬= 𝒓
𝟒𝝅𝜺𝟎 𝒓𝟐
Area vector: A vector having same magnitude as that of area and directed normal to the surface area.

ds  ds nˆ
Electric Flux(B) : Electric flux through a surface is the total number of the lines of force crossing the surface
in a direction normal to the surface

dB  E.ds
Consider a surface area „S‟ in an electric field. Let dS be the small area element. Let „E‟ be the electric field
and  be the angle b/w electric field and area vector. The Electric flux through the small area is
d  E ds cos 

15
If the surface is plane and the Electric field is uniform.

(a) If the surface is perpendicular to the electric field   0o 

E  E. A
E  EA cos 

(b) If the surface is parallel to the electric field   90o 

E  0
(c) If the surface is neither perpendicular nor parallel to the electric field makes an angle θ with the field
E  EA cos 

(d) Surface of arbitrary shape E   E.ds

Electric flux density (E): The electric flux per unit normal area around a point is called Electric flux density.
The relation between electric flux and electric flux density is  =  E dS cos , when  = 0
 =  E dS
Note:1. The SI unit of electric flux is N C-1 m2
2. The dimensional formula is [M L3 T-3 A-1]

16
Gauss Law: It states that “the total electric flux through any closed surface is equal to 1/o times the net
charge enclosed in that surface.
OR
The surface integral of electric field E produced by any source over any closed surface S enclosing the charge
is 1 / ε0 times the charge enclosed inside the closed surface.
 =  E dS
Explanation :

Consider a point charge +q at the centre of the sphere of radius r. Let P be a point on the surface of the sphere
and E is the electric field at that point.
Let ds be a small area element around the point P the electric flux through ds
d  Eds cos 
 0
d  Eds  (1)
1 q
W.K.T E  (2)
4 0 r 2
Sub (2) in (1)
1 q
 d   4 0 r2
ds

1q
4 0 r 2 
 ds

 ds  total area  4 r 2
1 q
E  4 r 2
4 0 r 2

q
E 
0
Limitation of Gauss Theorem:
It is applicable only for closed surface enclosing certain amount of charges.
Note : 1. The surface to which gauss is applicable is called Gaussian surface.
2. All Gaussian surfaces are closed surfaces but all closed surfaces are not Gaussian surface.
3. If a closed surface encloses electric dipole, the electric flux through it is zero. Therefore the total
charge is zero

17
Application of Gauss theorem:
Expression for Electric field due to an infinitely long uniformly charged straight wire:
Consider an infinitely long straight wire with uniform linear charge density „ ‟.
Let us imagine a cylindrical Gaussian surface.
The total electric flux on that surface is

𝝓= 𝑬 𝚫𝑺 + 𝑬 𝚫𝑺 + 𝑬𝚫𝑺
𝑺𝟏 𝑺𝟐 𝑺𝟑

𝝓=𝑬 𝚫𝑺𝒄𝒐𝒔𝟎 + 𝑬 𝚫𝑺𝒄𝒐𝒔𝟗𝟎 + 𝑬 𝚫𝑺 𝒄𝒐𝒔𝟗𝟎

𝑺𝟏 𝑺𝟐 𝑺𝟑

𝜙=𝐸 Δ𝑆
𝑆
Flux through the curved cylindrical part of the surface is,
𝜙 = 𝐸 2𝜋𝑟𝐿 → 1
From Gauss‟s law
𝟏 𝝀𝑳
𝝓= 𝒒= →𝟐
𝜺𝟎 𝜺𝟎
From eqn 1 and 2
𝝀𝑳
𝐸 2𝜋𝑟𝐿 = 𝜺
𝟎
𝝀
𝑬 = 𝒏
𝟐𝝅𝒓𝜺𝟎
Expression for the electric field due to uniformly charged infinite plane sheet:

Consider a uniformly charged infinite plane sheet. Let „σ‟ be the surface charge density and A be the area of end faces.
The total electric flux is given by
  EA  EA
 2 EA  (1)
From gauss law
1
 q  (2)
0
From eqn 1 and eqn 2
1
2 EA  q
0
q q
E 
2 A 0 A

E nˆ
2 0

Electric field is independent of „x‟. E remains uniform

18
Expression for electric intensity at a point near a charge spherical shell:
Consider a positively charged spherical conductor of radius „R‟ kept in air „q‟ be the magnitude of charge
on the surface of the conductor. Let „P‟ be a point outside the spherical conductor at a distance „r‟ from the
centre „O‟. Draw a imaginary sphere of radius „r‟. Since charges are uniformly distributed, electric intensity
is constant at all the points and also to the Gaussian surface.
The total Electric flux through the Gaussian surface is

E   E ds cos  (  0)

E  E  ds

 ds  4 r
2

E  E  4 r 2   (1)

1
From Gauss‟s law E  q  (2)
0

E  4 r 2  
1
From (1) and (2) q
0
1 q q
E 
 0 4 r 2 A

q

4 r 2
Note: If the point P is just on the surface of a charged spherical conductors then  r  R 

1 q
E
 0 4 r 2

E
0

Electric field inside the shell:

If the point „P‟ is inside the charged spherical conductor, the electric field is zero. Because the charges reside
only on the surface of the conductor
E  4 r 2  0
i.e., E  0
  0

19