Properties of Electric Charge

(i) There are two kinds of charge. As a matter of convention, charge of a proton is taken as positive and that of an electron is taken as negative.

(ii) Like charges repel each other, while unlike charges attract each other.

(iii) Charge is quantized. Charge on any object is integral multiple of electronic charge. i.e.; q = ± ne, where n = 1, 2 ......., e = 1.6 x 10^-19 coulomb (SI unit of change is coulomb).

(iv) Charge is conserved. In all types of phenomena, for an isolated system, total charge is conserved.

Units :

SI unit of charge is coulomb (C) There are other units of charge. Electromagnetic unit (emu) : 1 emu =10 C   Electro static unit (esu) : 1 esu =  x 10-9 C or, 1 C = 3 x 109 esu The electrostatic unit of charge is also called stat coulomb or Frankline (Fr).

Coulomb's Law

It states that the force of attraction or repulsion between two point charges, q₁ and q₂ at rest placed in "free space" separated by a distance r, is directly proportional to product of the charges and is inversely proportional to the square of distance between them i.e.

Here  is a constant, its value is 9 x 10⁹ N-m²/C²

e_o is called permittivity of free space and is equal to 8.85 x 10^-12 C²/N-m².

Permittivity is electrical property of a medium or that of free space.

Note that the term "free space" means that there is no medium between the charges, not even air.

When the space around the two interacting charges is filled by a medium, the force experienced by each of them changes.

If the charges are placed in a medium, the force experienced by each charge is given by

Note that the above expression, " e_o " has been replaced by " e ". It is the absolute permittivity of medium.

Relative Permittivity or Dielectric Constant (e_r or K)

It represents the ratio of absolute permittivity of a medium and absolute permittivity of free space.

The range of dielectric constant K is between '1' and '¥'.

Using Coulomb's Law:

You have just read an expression for the force of interaction between the charges. Before you start using the above expression in different problems, you must learn the following points about the force. Carefully study each of them.

(i) The force between the two charged particles is a central force. Following two figures describe the difference between a central and non central force.

(ii) The force is attractive , when the charges have opposite nature, i.e, one of the charge is positive while the other is negative. See the following figure




The positive charge is attracted to right and negative charge is attracted to left.

(iii) The force is repulsive , when the charges have same nature, i.e.; either both are positive or both are negative.

Vector Form of Coulomb's Law

A charge q₁ is placed at A whose position vector is 

Another charge q₂ is placed at B whose position vector is , such that | AB | = r

The magnitude of force is given by,

Force on q2 due to q1, in vector form is given by

     where  ia a unit vector along line joining A and B, pointing from A to B.

Electric Field - Electrostatics

When a charged particle placed at a point of space, experiences a force, a field is said to be present in the space. This field which exerts force on a charge is called electric field.

Electric field at a point P near a charged object can be defined as

Where q_o is positive point charge (called test charge) and  is the force experienced by it when it is placed at P.

The magnitude of electric field is   , where F = |  |. The direction of electric field is the direction of forced experienced by the test charge.

Thus it follows that when a charge q is placed in electric field, it experiences a force 

Units: SI unit of electric field is N/C. (or V/m)

Electric Field Due to a Point Charge 

A point charge q is placed at O. To define electric field at a point P, we place at test charge q_o and find the force acting on it.

Magnitude of electric field  

This direction of electric field is along the line OP away from O.

Electric field vectors due to point charge

Superposition principle 
It states that at a given point, the electric field due to separate charge distributions simply add up vectorially.

We can find the net or resultant electric field due to more than one point charge with the aid to principle of superposition. If we place a test charge qo near n point charges, the net force acting on it is.

     (whereis force on test charge due to i^th charge)

Electric Lines of Force or Electric Field Lines

The electric lines of force are purely a geometrical construction, which help us to visualise the nature of electric field in the region. They have the following characteristics:

1. The tangent to electric lines of force at any point gives the direction of electric field at the point.
    
2. In free space, they are continuous curves which emerge from positive charge and terminate at negative charge.
    
3. They do not intersect each other, If they do so, then it would mean two directions of electric field at the point of intersection, which is not possible.
    
4. The density of field lines represent strength of electric field in the region.

Field Lines in Some Cases

(a) Positive point charge

Field lines have spherical symmetry

(b) Negative point charge

Field lines have spherical symmetry

(c) Two similar charges of equal magnitude

(d) Two similar charges of unequal magnitudes

(e) Two dissimilar charges of equal magnitudes

Flux of Electric Field

In general, flux is used to represent flow, like water flowing through a pipe. Here flux is used to represent the number of fields lines that cross through a surface.

Flux of electric field through an infinitesimally small surface is defined by the relation

Total flux through a given surface area is



Units: Sl unit of flux is N-m²/C or V-m (volt metre)

Consider the following three cases of uniform field. These are helpful in explaining, how flux can be calculated

It is to be noted that while calculating flux through an open surface  can be taken in any direction normal to the surface. But for closed surface, outward normal is the positive direction of area vector (d or ). Thus while calculating flux through a closed surface, outward flux is taken as positive and inwards flux is taken as negative.

Gauss Law

The flux of net electric through a closed surface equals    times the charge enclosed by the surface.

Mathematically



Here  represents the net electric field vector at the surface due to all the charges (inside and outside).

Calculating Electric Field Using Gauss Law

Electric field due to a point charge

First, we shall look for the symmetry of the field. Clearly the field is spherically symmetric. If we enclose the charge in a sphere of radius R, the magnitude of electric field will be same at any point on the surface.



By Gauss law, [ q_enc = q]

[ q_enc = q]

 This is the same result as given by Coulomb's law.