Electrostatic Theory ~ Fundamentals of Physics

Electrostatics

Branch of science that deals with the study of forces, fields, and potentials arising from the static charges

Electric Charge

In 600 B.C., the Greek Philosopher Thales observed that amber, when rubbed with wool, acquires the property of attracting objects such as small bits of paper, dry leaves, dust particles, etc.
This kind of electricity developed on objects, when they are rubbed with each other, is called frictional electricity.
The American scientist Benjamin Franklin introduced the concept of positive and negative charges in order to distinguish the two kinds of charges developed on different objects when they are rubbed with each other.

In the table given below, if an object in the first column is rubbed against the object given in second column, then the object in the first column will acquire positive charge while that in second column will acquire negative charge.

Electric charge − The additional property of protons and electrons, which gives rise to electric force between them, is called electric charge.

Electric charge is a scalar quantity. A proton possesses positive charge while an electron possesses an equal negative charge (where e = 1.6 × 10⁻¹⁹ coulomb).

Like charges repel each other whereas unlike charges attract each other.
A simple apparatus used to detect charge on a body is the gold-leaf electroscope.

Conductors and Insulators
Conductors

The substances which allow electricity to pass through them easily are called conductors.

Example − All the metals are good conductors.

Conductors have electrons that can move freely inside the material.
When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor.
When a charged body is brought in contact with the earth, all the excess charge on the body disappears by causing a momentary current to pass to the ground through the connecting conductor (such as our body). This process is known as earthing.

Insulators

The substances which do not allow electricity to pass through them easily are called insulators.

Most of the non-metals such as porcelain, wood, nylon, etc. are examples of insulator.
If some charge is put on an insulator, then it stays at the same place.

Charging By Induction

A conductor may be charged permanently by induction in the following steps.

Step I

To charge a conductor AB negatively by induction, bring a positively charged glass rod close to it. The end A of the conductor becomes negatively charged while the far end B becomes positively charged. It happens so because when positively charged glass rod is brought near the conductor AB, it attracts the free electrons present in the conductor towards it. As a result, the electron accumulates at the near end A and therefore, this end becomes negatively charged and end B becomes deficient of electrons and acquires positive charge.

Step II

The conductor is now connected to the earth. The positive charges induced will disappear. The negative induced charge on end A of the conductor remains bound to it due to the attractive forces exerted by the positive glass rod.

Step III

The conductor is disconnected from the earth keeping the glass rod still in its position. End A of the conductor continues to hold the negative induced charge.

Step IV

Finally, when the glass rod is removed, the negative induced charge on the near end spreads uniformly over the whole conductor.

Basic Properties of Electric Charges

Additive nature of charges − The total electric charge on an object is equal to the algebraic sum of all the electric charges distributed on the different parts of the object. Ifq₁, q₂, q₃, … are electric charges present on different parts of an object, then total electric charge on the object, q = q₁ + q₂ + q₃ + …

Charge is conserved − When an isolated system consists of many charged bodies within it, due to interaction among these bodies, charges may get redistributed. However, it is found that the total charge of the isolated system is always conserved.
Quantization of charge − All observable charges are always some integral multiple of elementary charge, e (= ± 1.6 × 10⁻¹⁹ C). This is known as quantization of charge.

Coulomb’s Law

Two point charges attract or repel each other with a force which is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them.

Where,

[In SI, when the two charges are located in vacuum]

− Absolute permittivity of free space = 8.854 × 10⁻¹² C² N⁻¹ m⁻²

We can write equation (i) as

The force between two charges q₁ and q₂ located at a distance r in a medium may be expressed as

Where

− Absolute permittivity of the medium

The ratio

is denoted by ε_r, which is called relative permittivity of the medium with respect to vacuum. It is also denoted by k, called dielectric constant of the medium.

ε = kε₀

Coulomb’s Law in Vector Form

Consider two like charges q₁ and q₂ present at points A and B in vacuum at a distance r apart.
According to Coulomb’s law, the magnitude of force on charge q₁ due to q₂ (or on charge q₂ due to q₁) is given by,

Let

− Unit vector pointing from charge q₁ to q₂

− Unit vector pointing from charge q₂ to q₁

[

is along the direction of unit vector

] …(ii)

[

is along the direction of unit vector

] …(iii)

∴Equation (ii) becomes

On comparing equation (iii) with equation (iv), we obtain

Forces between Multiple Charges
Principle of superposition − Force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time. The individual forces are unaffected due to the presence of other charges.

Consider that n point charges q₁, q₂, q₃, … q_n are distributed in space in a discrete manner. The charges are interacting with each other. Let the charges q₂, q₃, … q_n exert forces

on charge q₁. Then, according to principle of superposition, the total force on charge q₁ is given by,

If the distance between the charges q₁ and q₂ is denoted as r₁₂; and

is unit vector from charge q₂ to q₁, then

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

Substituting these in equation (i),

Electric Field − It is the space around a charge, in which any other charge experiences an electrostatic force.

Electric Field Intensity − The electric field intensity at a point due to a source charge is defined as the force experienced per unit positive test charge placed at that point without disturbing the source charge.

Where,

→ Electric field intensity

Force experienced by the test charge q₀

Its SI unit is NC⁻¹.

Electric Field Due To a Point Charge

We have to find electric field at point P due to point charge +q placed at the origin such that

To find the same, place a vanishingly small positive test charge q₀ at point P.

According to Coulomb’s law, force on the test charge q₀ due to charge q is

is the electric field at point P, then

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

Representation of Electric Field

Electric Field Due To a System of Charges

Consider that n point charges q₁, q₂, q₃, … q_n exert forces

on the test charge placed at origin O.

Let

be force due to i^th charge q_i on q₀. Then,

Where, r_i is the distance of the test charge q₀ from q_i

The electric field at the observation point P is given by,

is the electric field at point P due to the system of charges, then by principal of superposition of electric fields,

Using equation (i), we obtain

Electric Field Lines

An electric line of force is the path along which a unit positive charge would move, if it is free to do so.

Properties of Electric Lines of Force

These start from the positive charge and end at the negative charge.
They always originate or terminate at right angles to the surface of the charge.
They can never intersect each other because it will mean that at that particular point, electric field has two directions. It is not possible.
They do not pass through a conductor.
They contract longitudinally.
They exert a lateral pressure on each other.

Representation of Electric Field Lines

Field lines in case of isolated point charges

Field lines in case of a system of two charges

Continuous Charge Distribution

Linear charge density − When charge is distributed along a line, the charge distribution is called linear.

Where,

λ → Linear charge density

q → Charge distributed along a line

L → Length of the rod

Surface charge density

Where,

σ → Surface charge density

q → Charge distributed on area A

Volume charge density

Where,

δ → Volume charge density

Electric Dipole − System of two equal and opposite charges separated by a certain small distance.

Electric Dipole Moment − It is a vector quantity, with magnitude equal to the product of either of the charges and the length of the electric dipole

Its direction is from the negative charge to the positive charge.

Electric Field on Axial Line of an Electric Dipole

Let P be at distance r from the centre of the dipole on the side of charge q. Then,

Where,

is the unit vector along the dipole axis (from − q to q). Also,

The total field at P is

For r >> a

Electric Field for Points on the Equatorial Plane

The magnitudes of the electric field due to the two charges +q and −q are given by,

The directions of E₊_q and E₋_q are as shown in the figure. The components normal to the dipole axis cancel away. The components along the dipole axis add up.

∴ Total electric field

[Negative sign shows that field is opposite to

]

At large distances (r >> a), this reduces to

Dipole in a Uniform External Field

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

making an angle θ with electric field.

Force on charge −q at

(opposite to

)

Force on charge +q at

(along

)

Electric dipole is under the action of two equal and unlike parallel forces, which give rise to a torque on the dipole.

τ = Force × Perpendicular distance between the two forces

τ = qE (AN) = qE (2a sin θ)

τ = q(2a) E sinθ

τ = pE sinθ

→ Volume of the conductor

Electric Flux Gauss Law

The electric flux, through a surface, held inside an electric field represents the total number of electric lines of force crossing the surface in a direction normal to the surface.

Electric flux is a scalar quantity and is denoted by Φ.

SI unit − Nm² C⁻¹

Gauss Theorem

It states that the total electric flux through a closed surface enclosing a charge is equal to

times the magnitude of the charge enclosed.

However,

∴Gauss theorem may be expressed as

Proof

Consider that a point electric charge q is situated at the centre of a sphere of radius ‘a’.

According to Coulomb’s law,

Where,

is unit vector along the line OP

The electric flux through area element

is given by,

Therefore, electric flux through the closed surface of the sphere,

It proves the Gauss theorem in electrostatics.

q → Charge on conductor

Applications of Gauss Law

Electric Field Due To A Line Charge

Consider a thin infinitely long straight line charge of linear charge density λ.

Let P be the point at a distance ‘a’ from the line. To find electric field at point P, draw a cylindrical surface of radius ‘a’ and length l.

If E is the magnitude of electric field at point P, then electric flux through the Gaussian surface is given by,

Φ = E × Area of the curved surface of a cylinder of radius r and length l

Because electric lines of force are parallel to end faces (circular caps) of the cylinder, there is no component of field along the normal to the end faces.

Φ = E × 2πal … (i)

According to Gauss theorem, we have

From equations (i) and (ii), we obtain

Electric Field Due To An Infinite Plane Sheet Of Charge

Consider an infinite thin plane sheet of positive charge having a uniform surface charge density σon both sides of the sheet. Let P be the point at a distance ‘a’ from the sheet at which electric field is required. Draw a Gaussian cylinder of area of cross-section A through point P.

The electric flux crossing through the Gaussian surface is given by,

Φ = E × Area of the circular caps of the cylinder

Since electric lines of force are parallel to the curved surface of the cylinder, the flux due to electric field of the plane sheet of charge passes only through the two circular caps of the cylinder.

Φ = E × 2A … (i)

According to Gauss theorem, we have

Here, the charge enclosed by the Gaussian surface,

q = σA

From equations (i) and (ii), we obtain

Electric Field Due To A Uniformly Charged Thin Spherical Shell

When point P lies outside the spherical shell

Suppose that we have to calculate electric field at the point P at a distance r (r > R) from its centre. Draw the Gaussian surface through point P so as to enclose the charged spherical shell. The Gaussian surface is a spherical shell of radius r and centre O.

Let