Charge And Force

The study of charges at rest is called static electricity or electrostatics.
Charges are produces due to actual transfer of electrons. So charge is defined as excess or deficit of electrons. The unit of charge is coulomb(C) and $1 C = 3 * 1 0^{9} s t a t co u l o mb$ .
Positive charge means deficiency of electrons while negative charge means excess of electrons. So, during charge transfer, there is change in mass too(due to transfer of electrons). Hence, the mass of negatively charged body is greater than that of positively charged body.
The properties of charges are:

a. Quantization of charge:

Electric charge can only exist as an integral multiple of charge on an electron i.e

$q = n e$

b. Conservation of charge:

In an isolated system, total electric charge always remains constant.

c. Like charges repel each other while unlike charged attract each other.

d. A charge at rest produces only electric field around itself. A charge having uniform motion (unaccelerated) produces electric as well as magnetic field around it. While a charge having non-uniform motion (accelerated) emits electromagnetic radiation also in addition to producing electric and magnetic fields.

A body can be charged by:

a. Friction

b. Conduction

c. Induction

Friction: It is usually done by rubbing a substance over other. For example: When a glass rod is rubbed with fur, positive charge is developed in the glass rod because free electrons are transferred from glass rod to the fur and when an ebonite rod is rubbed with silk, negative charge is developed in the ebonite rod because free electrons are transferred from silk to ebonite rod.
Conduction: The charge is transferred from one body to another when they are brought in contact with one another. During conduction the same charge is created in a neutral object. For example: electrons will transfer from a negative object to a neutral object making it negative.
Induction: It is a charging method that charges an object without actually touching the object to any other charged object. The charging by induction process is where the charged particle is held near an uncharged conductive material that is grounded on a neutrally charged material. It is also known as temporary electrification. The inducing charge produces induced charge in the neutral body. The relation between inducing and induced charge is:

$q_{in d u ce d} = - q_{in d u c in g} (1 - \frac{1}{E r})$

where Er is the dielectric constant and the '-' sign indicates the opposite nature of induction.

For air or vacuum, $E r = 1$ . Hence there is no induction in air.

For conductor, $E r = \infty$ . Hence the induced charge is always equal to inducing charge.

For insulator, $E r > 1$ . Hence the induced charge is always less than the inducing charge.

The types of charge carriers are:

a. Conductors: They easily allow electricity to pass through them. For example: Metals.

b. Insulators: They do not allow electricity to pass through them. For example: Wood.

c. Semi-conductors: They lie in between conductors and insulators in their ability to conduct electricity. For example: Silicon and Germanium.

Charge density:

a. Linear charge density: Charge per unit length on linear object is called linear charge density and it is denoted by $λ$ . Its unit is coulomb per meter.

$λ = \frac{q}{l}$

b. Surface charge density: Charge per unit area is known as surface charge density. Its unit is coulomb per square meter. It decreases with increase of radius of curvature of the body.

$σ = \frac{q}{A}$

c. Volume charge density: Charge per unit volume of any charged bulk matter is called volume charge density. Its unit is coulomb per cubic meter.

$ρ = \frac{q}{V}$

Coulomb's law: The force of attraction or repulsion between two point charge is directly proportional to the product of charge and inversely proportional to the square of the distance between them.

$F \propto q_{1} * q_{2}$

$F \propto \frac{1}{r ^{2}}$

Hence,

$F = \frac{q _{1} * q _{2}}{4 π E _{o} r ^{2}}$

where,

$\frac{1}{4 π E _{o}}$ = $9 * 1 0^{9}$ $N m^{2} C^{- 2}$ and $E_{o} = 8.85 * 1 0^{- 12} C^{2} N^{- 1} m^{- 2}$ is called permittivity of free space.

-> $F_{m e d i u m} = \frac{1}{4 π E} (\frac{q _{1} q _{2}}{r ^{2}}) = \frac{1}{4 π E _{o} E _{r}} (\frac{q _{1} q _{2}}{r ^{2}})$

Hence, the dielectric constant is given by:

$E_{r} = \frac{F _{ai r}}{F _{m e d i u m}} = \frac{E}{E _{r}}$

It has no units and no dimensions.

Gold Leaf electroscope: It is used to detect the presence of charge and also find the nature of charge in the body. If similar charged body is brought near, the gold leaves diverge and vice versa.

Important tips:
For a system to be in equilibrium, the net force should be zero.
When two identical conductors having charge $q_{1}$ and $q_{2}$ $q_{1} an d q_{2}$ are put in contact then separated, each conductor has charge $= \frac{q _{1} + q _{2}}{2}$