## What is called Electrostatic Force?

*Static charge of a body is the fundamental physical property of a matter that causes to feel a force of attraction or repulsion due to another charged body. This force of attraction or repulsion between charged bodies is called electrostatic force.*

Magnitude of electrostatic force was first formulated by Coulomb. Hence, it is also called Coulomb’s force. The force of attraction between two *static charges* is measured by using a device called **Torsion balance.**

__Coulomb’s law of Electrostatic Force__

Coulomb establishes an expression for electrostatic force of attraction or repulsion between static charges. Coulombs law states that –

The magnitude of force of attraction or repulsion between two point charges is –

- Directly proportional to the product of the magnitude of charges.
- Inversely proportional to the square of the distance between the charges.

Consider about two point charges ( q_1 ) and ( q_2 ) which are kept at a distance of ( r ) as shown in figure.

Then, by Coulomb’s law –

- F \propto q_1 q_2
- F \propto \left ( \frac {1}{r^2} \right )

Therefore, force between two charges will be –

F \propto \left ( \frac { q_1 q_2 } { r^2 } \right )

Or, \quad F = k \left ( \frac { q_1 q_2 } { r^2 } \right )

The force ( F ) is called ** electrostatic force** or

**and ( k ) is a constant of proportionality called**

*Coulomb’s force*

*Coulomb’s constant.*The value of Coulomb’s constant ( k ) depends upon –

- The system of units used for charges and
- The nature i.e. permittivity of medium in which the charges are kept.

In SI system of units –

k = \left ( \frac {1}{4 \pi \epsilon} \right ) = \left ( \frac {1}{4 \pi \epsilon_0 \epsilon_r} \right )

Where, ( \epsilon ) is the *permittivity* of the medium, ( \epsilon_0 ) is the absolute permittivity of free space and ( \epsilon_r ) is the *relative permittivity* of the medium with respect to free space.

Therefore, \quad \epsilon = \left ( \epsilon_0 \times \epsilon_r \right )

The value of *absolute permittivity* of free space is –

\epsilon_0 = 8.854 \times 10^{-12} \ C^2 \ N^{-1} \ m^{-2}

__Characteristics of Electrostatic Force__

Electrostatic force has the following characteristics –

- It is a central force i.e. it acts along the line joining the centres of the charged particles.
- Electrostatic force is spherically symmetric because it is a function of ( r ) (distance between charges)
- Force is attractive between two unlike charges ( i.e one positive charge and other negative charge) and repulsive between two like charges ( i.e. both positive charges or both negative charges).
- Electrostatic force between two charges is not affected by the presence of other charges.
- These are
*conservative forces**.*Therefore,*work done*in moving a charge over a closed path in an*electric field*is zero.

__Charge Distribution__

Electrostatic force between bodies depend upon the distribution of charge on their bodies. Three types of distribution of charge is found in solid objects. These are –

- Linear charge distribution.
- Surface charge distribution.
- Volume charge distribution.

__Linear Charge Distribution__

*One-dimensional distribution of charge is known as linear charge distribution. *

This type of charge distribution is found in –

- Long wires.
- Cylindrical rods.
- Charged ring etc.

__Electrostatic Force in Linear Charge Distribution__

Consider about a straight wire AB of length ( l ) uniformly distributed with a total charge ( q ) as shown in figure.

Then ** linear charge density** on the wire will be –

\lambda = \left ( \frac {q}{l} \right )

Consider a small element of length dl of the wire. Total charge on this element will be –

dq = \lambda dl ……… (1)

Electrostatic force on a test charge q_0 placed at a point P at a distance ( r ) from the element is given by –

dF = \left ( \frac {1}{4 \pi \epsilon_0} \right ) \left ( \frac {dq \times q_0}{r^2} \right ) ……… (2)

Total force on ( q_0 ) due to charge on the whole wire will be obtained by integrating equation (2) for whole length.

Therefore, \quad F = \int\limits_{l} dF = \int\limits_{l} \left ( \frac {1}{4 \pi \epsilon_0} \right ) \left ( \frac {dq \times q_0}{r^2} \right )

Using equation (1) we have –

F = \int\limits_{l} \left ( \frac {1}{4 \pi \epsilon_0} \right ) \left ( \frac {\lambda dl \times q_0}{r^2} \right )

Therefore, \quad F = \left ( \frac {\lambda q_0}{4 \pi \epsilon_0} \right ) \int\limits_{l} \left ( \frac { 1 }{r^2} \right ) dl ………. (3)

__Surface Charge Distribution__

*A two dimensional charge distributions is called surface charge distribution. *

Surface charge distribution is found in –

- Thin films.
- Plane sheets etc.

__Electrostatic Force in Surface Charge Distribution__

Consider a plane sheet of surface area ( S ) on which charge ( q ) is uniformly distributed as shown in figure.

Then, ** Surface charge density** on the sheet will be –

\sigma = \left ( \frac {q}{S} \right )

Consider a small element of area ( dS ) of the sheet. Total charge on this element will be –

dq = \sigma dS …….. (4)

Electrostatic force on a test charge ( q_0 ) placed at a point P at a distance ( r ) from the element is given by –

dF = \left ( \frac {1}{4 \pi \epsilon_0} \right ) \left ( \frac {dq \times q_0}{r^2} \right ) ………. (5)

Therefore, total force on charge ( q_0 ) due to charge on the whole sheet is obtained by integrating equation (5) for whole surface.

Therefore, \quad F = \int\limits_{S} dF = \int\limits_{S} \left ( \frac {1}{4 \pi \epsilon_0} \right ) \left ( \frac {dq \times q_0}{r^2} \right )

Using equation (4) we have –

F = \int\limits_{S} \left ( \frac {1}{4 \pi \epsilon_0} \right ) \left ( \frac {\sigma dS \times q_0}{r^2} \right )

Therefore, \quad F = \left ( \frac {\sigma q_0}{4 \pi \epsilon_0} \right ) \int\limits_{S} \left ( \frac { 1 }{r^2} \right ) dS ………. (6)

__Volume Charge Distribution__

*Volume charge distribution is a three dimensional charge distributions.*

This type of charge distribution is found in –

- Solid objects.
- Charge on a bulb.
- Sphere etc.

#### Electrostatic Force in Volume Charge Distribution –

Consider about a solid object of volume ( V ) on which charge ( q ) is uniformly distributed as shown in figure.

Then, ** volume charge density** on the object will be –

\rho = \left ( \frac {q}{V} \right )

Consider a small element of volume ( dV ) of the object. Total charge on this element will be –

dq = \rho dV ……… (7)

Electrostatic force on a test charge ( q_0 ) placed at a point P at a distance ( r ) from the element is given by –

dF = \left ( \frac {1}{4 \pi \epsilon_0} \right ) \left ( \frac {dq \times q_0}{r^2} \right ) ………. (8)

Therefore, total force on ( q_0 ) due to charge on the whole object is obtained by integrating the equation (8) for whole object.

F = \int\limits_{V} dF = \int\limits_{V} \left ( \frac {1}{4 \pi \epsilon_0} \right ) \left ( \frac {dq \times q_0}{r^2} \right )

Using equation (7) we will get –

F = \int\limits_{V} \left ( \frac {1}{4 \pi \epsilon_0} \right ) \left ( \frac {\rho dV \times q_0}{r^2} \right )

Therefore, \quad F = \left ( \frac {\rho q_0}{4 \pi \epsilon_0} \right ) \int\limits_{V} \left ( \frac { 1 }{r^2} \right ) dV ………. (9)

See numerical problems based on this article.