# Electrostatics

1. Introduction
2. Electric Charge
3. Coulomb’s Law of Electrostatics
4. Electric Field
5. Electric Field Lines
6. Conservative Force
7. Potential Energy
8. Gauss’s Theorem
9. Electric Dipole
10. Capacitor

### What is Electrostatics

Electrostatics is the study of stationary electric charges. Electric charge is the charge attached to matter that causes it to produce and experience a magnetic and electric effect. A plastic rod will attract pieces of paper when stroked with fur, and a glass rod will do the same when rubbed with silk, indicating that both rods are electrically charged.

Electrostatics is a branch of physics that deals with the study of electromagnetic phenomena where electric charges are at rest, i.e., where no moving charges exist after a static equilibrium has been established. The electrostatic phenomenon in physics deals with the characteristics of fixed or slowly moving electric charges. Additionally, Coulomb's Law defines this phenomena since it results from the forces that electric charges exert on one another. As a result, understanding electric charge and Coulomb's Law is necessary in order to comprehend the concept of electrostatic.

### What is Electric Charge?

Electric Charge is the property associated with matter due to which it produces and experiences electrical and magnetic effects. The excess or deficiency of electrons in a body gives the concept of charge.

The fundamental quality of subatomic particles that enables them to feel a force when exposed to an electromagnetic field is known as electric charge, also known as charge or electrostatic charge. Electric charges will be either positive or negative. Positive charges are carried by protons, while negative charges are carried by electrons. An object is considered neutral if its net charge is zero, i.e., neither positive nor negative. The Coulomb is the SI unit of charge.

SI unit of charge : Coulomb(C)
Dimension: ampere × second = [AT] = [M0L0T1I1]
C.G.S unit of charge = electrostatic unit = esu
1 coulomb = 3 × 109 esu of charge

### Positively Charged Particles

In this type of particle, numbers of protons are larger than the numbers of electrons.

### Negatively Charged Particles

In this type of particle, electrons are larger in number than that of protons.

## Properties of Charge

• Charge is a Scalar Quantity: It adds algebrically and represents excess, or deficiency of electrons.

• Charge is Transferable: Charging a body implies transfer of charge (electrons) from one body to another. Positively charged body means loss of electrons, i.e. deficiency of electrons. Negatively charged body means excess of electrons. This also shows that mass of a negatively charged body > mass of a positively charged identical body.

• Charge is Conserved: In an isolated system, total charge (sum of positive and negative) remains constant whatever change takes place in that system.

• Charge is Quantized: Charge on any body always exists in integral multiples of a fundamental unit of electric charge. This unit is equal to the magnitude of charge on electron (1e = 1.6 × 10-19 coulomb). So charge on anybody Q = ± ne, where n is an integer and e is the charge of the electron. Millikan's oil drop experiment proved the quantization of charge or atomicity of charge.
Recently a new particle has been discovered called ‘Quark’. It contains charge ± e / 3, ± 2e / 3. [The protons and neutrons are combination of other entities called quarks, which have charges 1 / 3 e. However, isolated quarks have not been observed, so, quantum of charge is still e. ]

• Charge is Relativistically Invariant:: This means that charge is independent of frame of reference, i.e., charge on a body does not change whatever be its speed. This property is worth mentioning as in contrast to charge, the mass of a body depends on its speed and increases with increase in speed.

• Like point charges repel each other while unlike point charges attract each other, which means ++ repels , -- repels, +- attracts and -+ attracts.

• Charge is always associated with mass, i.e., charge can not exist without mass though mass can exist without charge. The particle such as photon or neutrino which have no (rest) mass can never have a charge.

• A charge at rest produces only electric field around itself; a charge having uniform motion produces electric as well as magnetic field around itself while a charge having accelerated motion emits electromagnetic radiation.

• Ex-1: How many electrons are there in one coulomb of negative charge?

Sol: The negative charge is due to the presence of excess electrons, since they carry negative charge. Because an electron has a charge whose magnitude is e = 1.6 × 10-19 C, the number of electrons is equal to the charge q divided by the charge e on each electron. Therefore, the number n of electrons is n = q/e = 1/1.6 × 10-19=6.25 × 1018

## METHODS OF CHARGING

• Friction:If we rub one body with other body, electrons are transferred from one body to the other.

• Induction:If a charged body is brought near a metallic neutral body, the charged body will attract opposite charge and repel similar charge present in the neutral body. As a result of this one side of the neutral body becomes negative while the other positive, this process is called 'electrostatic induction'.
Charging a body by induction method includes these foue steps:

• Conduction:The process of transfer of charge by contact of two bodies is known as conduction. If a charged body is put in contact with uncharged body, the uncharged body becomes charged due to transfer of electrons from one body to the other.

The charged body loses some of its charge (which is equal to the charge gained by the uncharged body)
The charge gained by the uncharged body is always lesser than initial charge present on the charged body.

• To charge the bodies through friction one of them has to be an insultator.

### Conductors and Insulators

Conductors are material that allow the easy passage of electric charges. The most common conductors are all metals.

Insulators are substances that restrict the easy movement of electric charges because they possess strongly bonded electrons that are immobile.The most common examples of insulators are paper and rubber.

## Coulomb’s Law of Electrostatics

The electrostaticforce of interaction between two static point electric charges is directly proportional to the product of the charges, inversely proportional to the square of the distance between them and acts along the straight line joining the two charges. Force of electrostatic interaction depends on the nature of medium between the charges. If two points charges q1 and q2 separated by a distance r. Let F be the electrostatic force between these two charges.

According to Coulomb's law: F = 1 / 4π εo q1q2 / r2
where q1, q2 are magnitude of point charges, r is the distance between them and εo is permittivity of free space.
In examples and problems we will often use the approximate value,
1 / 4πεo = 9 × 109 N-m2/C2
The value of εo is 8.85 × 10-12 C2 / N-mC2.
If there is another medium between the point charges except air or vacuum, then εo is replaced by εoK or εoεr or ε.
where K or εr is called dielectric constant or relative permittivity of the medium.
K = εr = ε / εo
where, ε = permittivity of the medium.
For air or vacuum, K = 1
For water, K = 81
For metals, K = ∞

### Coulomb’s Law in Vector Form

Force on q2 due to q1,

The above equations give the Coulomb’s law in vector form.

• Head of r points at that position where force has to be calculated.

• r2 & r1 depend on origin but r does not.

• q1 due to q1 should be put along with sign.

• Force on q1 due to q2 = – Force on q2 due to q1

F12 = – F21

F12 = Kq1q2 . r1 – r2 / |r1 – r2|3

The forces due to two point charges are parallel to the line joining point charges; such forces are called central forces and electrostatic forces are conservative forces.

## Electric Field

The space in the surrounding of any charge in which its influence can be experienced by other charges is called electric field.

An electric field is a field that forms around an electrically charged body and acts as a force field on nearby charged items.

Electric Field is calculated by the term called electric field density. For instance, if you place a positive (+) unit charge close to a positively charged body, a repulsive force will occur. The resulted force will make the unit charge to move away from the body. The imaginary line over which this unit charge will move is termed as the line of forces.

Similarly, if you place a positive (+) unit charge in the field around a negatively charged object, it will experience a force of attraction. The resulted force will compel the unit positive charge to come closer to the negatively charged object. In this case, the line through which the unit charge moves is the line of forces.

Coulomb's law states that the force with which like charges repel each other and opposite charges attract each other is proportional to the product of the charges and inversely proportional to the square of the distance between them.

We show charge with ‘q’ or ‘Q’ and, the smallest unit charges are 1.6021x10⁻¹⁹ Coulomb(C) ie magnitude of charge of electron or proton.An electron and a proton have the same amount of charge.

## Electric Field Lines

Electric field lines(Electric Lines of Force) are an imaginary line or curve and a great way of visualizing electric fields. They were first introduced by Michael Faraday.

### Properties of Electric Field Lines

• The electric field lines are continous curves in an electric field starting from a positively charged body and ending on a negatively charged body.

• The tangent to the curve at any point gives the direction of the electric field intensity at that point.

• Crowded lines represent strong field while distant lines weak field.

• Electric lines of force never intersect since if they cross at a point, electric field intensity at the point will have two directions, which is not possible.

• The number of electric lines of force that originate from or terminate on a charge is proportional to the magnitude of the charge.

• If electric field lines are equidistant straight lines, the field is uniform.

• Electric field lines enter or exit a charged surface in a normal manner.

• It is not possible for electric field lines to go through a conductor. As such, inside a conductor, the electric field is always equal to zero.

• Electric field lines do not pass but leave or end on a charged conductor normally.

• ### Electric Field Intensity(E)

The electrostatic force acting per unit positive charge on a point in electric field is called electric field intensity at that point.
Electric field intensity E =

Its SI unit is NC-1 or Vim and its dimension is [MLT-3 A-1].
It is a vector quantity and its direction is in the direction of electrostatic force acting on positive charge.
Electric field intensity due to a point charge q at a distance r is given by
E = 1 / 4π εo q / r2

## Conservative Force

A force is conservative if work done by or against the force in moving a body depends only on the initial and final positions of the body and not on the nature of path followed between the initial and final positions.

Examples of Conservative Forces:

• Gravitational Force (Between two Masses)

• Electrostatic force (Between two Charges)

• Elastic force (Stretched or Compressed Spring)

• Magnetic force (Between two Magnetic Poles)

• Forces acting along the line joining the centres of two bodies are called central forces. Gravitational force and Electrosatic forces are two important examples of central forces. Central forces are conservative forces.

Properties of Conservative Forces:

• Work done by or against a conservative force depends only on the initial and final positions of the body.

• Work done by or against a conservative force in a round trip is zero. If a body moves under the action of a force that does no total work during any round trip, then the force is conservative; otherwise it is non-conservative.

• Work done by or against a conservative force does not depend upon the nature of the path between initial and final positions of the body. If the work done by a force in moving a body from an initial location to a final location is independent of the path taken between the two points, then the force is conservative.

• The work done by a conservative force is completely recoverable. Complete recoverability is an important aspect of the work of a conservative force.

• ### Non-Conservative Force

A force is said to be non-conservative if work done by or against the force in moving a body depends upon the path between the initial and final positions. The frictional forces are non-conservative forces. This is because the work done against friction depends on the length of the path along which a body is moved. It does not depend only on the initial and final positions. Note that the work done by fricitional force in a round trip is not zero. The velocity-dependent forces such as air resistance, viscous force, magnetic force etc., are non conservative forces

## Potential Energy (U):

Potential energy of a system of particles is defined only in conservative fields. As electric field is also conservative, we define potential energy in it.
Some important points regarding potential energy in electric fields:

• Doing work means supply of energy.

• Energy can neither be transferred nor be transformed into any other form without doing work.

• Kinetic energy implies utilization of energy where as potential energy implies storage of energy

• Whenever work is done on a system of bodies, the supplied energy to the system is either used in form of KE of its particles or it will be stored in the system in some form, increases the potential energy of system.

• When all particles of a system are separated far apart by infinite distance there will be no interaction between them. This state we take as reference of zero potential energy.

• Potential energy of a system of particles we define as the work done in assembling the system in a given configuration against the interaction forces of particles.

Electrostatic potential energy is defined in two ways.
• Interaction energy of charged particles of a system.
• Self energy of a charged object

• Electrostatic Interaction Energy
Electrostatic interaction energy of a system of charged particles is defined as the external work required to assemble the particles from infinity to the given configuration. When some charged particles are at infinite separation, their potential energy is taken zero as no interaction is there between them. When these charges are brought close to a given configuration, external work is required if the force between these particles is repulsive and energy is supplied to the system, hence final potential energy of system will be positive. If the force between the particle is attractive, work will be done by the system and final potential energy of system will be negative.

Interaction Energy of a System of Two Charged Particles :

Figure shows two +ve charges q1 and q2 separated by a distance r. The electrostatic interaction energy of this system can be given as work done in bringing q2 from infinity to the given separation from q1. If can be calculated as

## Electric Potential(V)

Electric potential is a scalar property of every point in the region of electric field. At a point in electric field, electric potential is defined as the interaction energy of a unit positive charge.
Electric potential at any point is equal to the work done per positive charge in carrying it from infinity to that point in electric field. Similarly we can define electric potential as "work done in bringing a unit positive charge from infinity to the given point against the electric forces."

Electric potential, V = W / q
Its SI unit is J / C or volt and its dimension is [ML2T-3A-1].
It is a scalar quantity.

Electric potential due to a point charge at a distance r is given by
v = 1 / 4π εo q / r

Properties :

• Potential is a scalar quantity, its value may be positive, negative or zero.

• S.I. Unit of potential is volt = joule/coulomb and its dimensional formula is [ML2T-3A-1].

• Electric potential at a point is also equal to the negative of the work done by the electric field in taking the point charge from reference point (i.e. infinity) to that point.

• Electric potential due to a positive charge is always positive and due to negative charge it is always negative except at infinity. (taking V=0)

• Potential decreases in the direction of electric field.

The rate of change of potential with distance in electric field is called potential gradient.
Potential gradient = dV / dr

Its unit is V / m.

Relation between potential gradient and electric field intensity is given by
E = – (dV / dr)

### Equipotential Surface

Equipotential surface is an imaginary surface joining the points of same potential in an electric field. So, we can say that the potential difference between any two points on an equipotential surface is zero. The electric lines of force at each point of an equipotential surface are normal to the surface.

• Equipotential surface can never cross each other.

• Electric field is always perpendicular to equipotential surface.

• Equipotential surface due to an isolated point charge is spherical.

• Equipotential surface are planer in an uniform electric field.

• Equipotential surface due to a line charge is cylindrical.

• The intensity of electric field along an equipotential surface is always zero.

• ### Electric Flux(φE)

Any group of electric lines of forces passing through a given surface, we call electric flux and it is denoted by φ.Electric flux over an area is equal to the total number of electric field lines crossing this area.
Electric flux through a small area element dS is given by

φE = E. dS
where E= electric field intensity and dS = area vector.
Its SI unit is N – m2C-1.

### Gauss’s Theorem

This law is the mathematical analysis of the relation between the electric flux from a closed surface and its enclosed charge. This law states that the total flux emerging out from a closed surface is equal to the product of sum of enclosed charge by the surface and the constant 1 / εo Mathematically Gauss’s law is written as:

If a charge q is placed at the centre of a cube, then
total electric flux linked with the whole cube = q / εo
electric flux linked with one face of the cube = q / 6 εo

### Electric Field at Any Point on the Axis of a Uniformly Charged Ring

A ring-shaped conductor with radius a carries total charge Q uniformly distributed around it. Let us calculate the electric field at a point P that lies on the axis of the ring at distance x from its centre.

Ex = 1 / 4π εo * xQ / (x2 + a2)3/2

The maximum value of electric field
Ex = 1 / 4π εo (2Q / 3√3R2)

### Electric Field due to a Charged Spherical Shell

(a) At an extreme point (r > R)
E = 1 / 4π εo q / r2
(b) At the surface of a shell (r = R)
E = 1 / 4π εo q / R2
(c) At an internal point (r < R)
E = 0

### Electric Potential due to a Charged Conducting Spherical Shell

(a) At an extreme point (r > R)
V = 1 / 4π εo q / r
(b) At the surface of a shell (r = R)
V = 1 / 4π εo q / R
(c) At an internal point (r < R)
V = 1 / 4π εo q / R
Therefore potential inside a charged conducting spherical shell equal to the potential at its surface.

### Electric Field and Potential due to a Charged Non-Conducting Sphere

At an extreme point, (r > R)

(v)

### Electric Field Intensity due to an Infinite Line Charge

E = 1 / 2 π εo λ / r
where λ is linear charge density and r is distance from the line charge.

### Electric Field Near an Infinite Plane Sheet of Charge

E = σ / 2 εo
where σ = surface charge density.
If infinite plane sheet has uniform thickness, then
E = σ / εo

## Electric Dipole

An electric dipole consists of two equal and opposite point charges separated by a very small distance. e.g., a molecule of HCL, a molecule of water etc.

Electric Dipole Moment p = q * 2 l
Its SI unit is ‘coulomb-metre’.
It is a vector quantity and its direction is from negative charge towards positive charge.

### On Axial Line

Electric field intensity E = 1 / 4 π εo * 2 pr / (r2 – a2)2
If r > > 2a, then E = 1 / 4 π εo * 2 p / r3
Electric potential V = 1 / 4 π εo * p / (r2 – a2)

Ifr > > 2a, then V = 1 / 4 π εo * p / r2

### On Equatorial Line

Electric field intensity E = 1 / 4 π εo * p / (r2 + a2)3 / 2

If r > > 2a, then E = 1 / 4 π εo * p / r3
Electric potential V = 0

### At any Point along a Line Making θ Angle with Axis

Electric field intensity E = 1 / 4 π εo * p √1 + 3 cos2 θ / r3

Electric potential V = 1 / 4 π εo * p cos θ / (r2 – a2 cos2 θ)
If r > > 2a, then V = 1 / 4 π εo * p cos θ / r2

### Torque

Torque acting on an electric dipole placed in uniform electric field is given by
τ = Ep sin θ or τ = p * E
When θ = 90°, then ‘τmax = Ep
When electric dipole is parallel to electric field, it is in stable equilibrium and when it is anti-parallel to electric field, it is in unstable equilibrium.

### Work Done

Work done is rotating an electric dipole in a uniform electric field from angle θ1 to θ2 is given by
W = Ep (cos θ1 – cos θ2)
If initially it is in the direction of electric field, then work done in rotating through an angle θ, W = Ep (1 – cos θ).

### Potential Energy

Potential energy of an electric dipole in a uniform electric field is given by U = – pE cos &theta.

### Dipole in Non-uniform Electric Field

When an electric dipole is placed in a non-uniform electric field, then a resultant force as well as a torque act on it.
Net force on electric dipole = (qE1 – qE2), along the direction of greater e ecmc field intensity.
Therefore electric dipole undergo rotational as well as linear motion.

### Potential Energy of Charge System

Two point charge system, contains charges q1 and q2 separated by a distance r is given by U = 1 / 4 π εo * q1q2 / r
Three point charge system

U = 1 / 4 π εo * [q1q2 / r1 + q2q3 / r2 + q3q2 / r3

### Important Points

When charge is given to a soap bubble its size gets increased.
In equilibrium for a charged soap bubble, pressure due to surface tension
= electric pressure due to charging
4T / r = σ2 / 2 εo
or 4T / r = 1 / 2 εo (q / 4 πr2)2
or q = 8 πr √2 εo rT
where, r is radius of soap bubble and T is surface tension of soap bubble.

### Behaviour of a Conductor in an Electrostatic Field

(i) Electric field at any point inside the conductor is zero.
(ii) Electric field at any point on the surface of charged conductor is directly proportional to the surface density of charge at that point, but electric potential does not depend upon the surface density of charge.
(iii) Electric potential at any point inside the conductor is constant and equal to potential.

### Electrostatic Shielding

The process of protecting certain field from external electric field is called, electrostatic shielding.
Electrostatic shielding is achieved by enclosing that region in a closed metallic chamber.

## Capacitor

A capacitor is a device which is used to store huge charge over it, without changing its dimensions.
When an earthed conductor is placed near a charged conductor, then it decreases its potential and therefore more charge can be stored over it.
A capacitor is a pair of two conductors of any shape, close to each other and have equal and opposite charges.
Capacitance of a conductor C = q / V
Its 81 unit is coulomb/volt or farad.
Its other units are 1 μ F = 10-6 F
1 μμ F = 1 pF = 10-12 F
Its dimensional formula is [M-1L-2T4A2].

### Capacitance of an Isolated Spherical Conductor

C = 4 π εo K R

For air K = 1

∴ C = 4 π εo R = R / 9 * 109

### Parallel Plate Capacitor

The parallel plate capacitor consists of two metal plates parallel to each other and separated by a distance d.
Capacitance C = KA εo / d
For air capacitor Co = A εo / d

When a dielectric slab is inserted between the plates partially, then its capacitance.

C = A εo / (d – t + t / K)
If a conducting (metal) slab is inserted between the plates, then
C = A εo / (d – t)
When more than one dielectric slabs are placed fully between the plates, then

The plates of a parallel plate capacitor attract each other with a force
F = Q2 / 2 A εo
When 9. dielectric slab is placed between the plates of a capacitor than charge induced on its side due to polarization of dielectric is
q’ = q (1 – 1 / k)

### In Series

Resultant capacitance = 1 / C = 1 / C1 + 1 / C2 + 1 / C3 + ….
In series charge is same on each capacitor, which is equal to the charge supplied by the source.
If V1, V2, V3,…. are potential differences across the plates of the capacitors then total voltage applied by the
source
V = V1 + V2 + V3 + ….

### In Parallel

Resultant capacitance C = C1 + C2 + C3 + ….
In parallel potential differences across the plates of each capacitor is same.
If q1, q2, q3 are charges on the plate of capacitors connected in parallel then total charge given by the source
q = q1 + q2 + q3 + ….
Electric potential energy of a charged conductor or a capacitor is given by,
U = 1 / 2 Vq = 1 / 2 CV2 = 1 / 2 q2 / C

### Redistribution of Charge

When two isolated charged conductors are connected to each other then charge is redistributed in the ratio of their capacitances.
Common potential V = q1 + q2 / C1 + C2 = C1V1 + C2V2 / C1 + C2
Energy loss = 1 / 2 C1C2 (V1 – V2)2 / (C1 + C2)
This energy is lost in the form of heat in connecting wires.
When n, small drops, each of capacitance C, charged to potential V with charge q, surface charge density σ and potential energy U coalesce to from a single drop.
Then for new drop
Total charge = nq
Total capacitance = nl/3C
Total potential = n2/3 V
Surface charge density = nl/3 σ
Total potential energy = n2/3 U

### Van-de-Graaff Generator

It is a device used to build up very high potential difference of the order of few million volt.

Its working is based on two points
(i) The action of sharp points (corona discharge)
(ii) Total charge given to a spherical shell resides on its outer surface.

### Lightning Conductor

When a charged cloud passes by a tall building, the charge on cloud passes to the earth through the building. This causes a damage to the building. Thus to protect the tall building lightning, the lightning conductors, (which are pointed metal roe passes over the charge on the clouds to earth, thus protecting building.

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