Electrostatics: Coulomb's Law, Electric Field, and Electric Potential

A micro-course in electrostatics covering Coulomb's law, electric field, and electric potential for high school students.

 

Electric Charge

 

Electric charge is one of the physical properties of matter. There are two types of electric charges in the universe: positive and negative electric charges.
Elementary electric charge: electric charge of a single proton, or the magnitude of the electric charge of a single electron is called elementary electric charge.
SI unit of electric charge: SI unit of electric charge is the Coulomb. One coulomb is equal to the electric charge of approximately 6.24 ×10¹⁸  protons ( 1 Coulomb ∼ 18|e|). So the electric charge of a single proton (or the magnitude of the charge of a single electron) is equal to approximately 1.6 × 10⁻¹⁹  Coulombs. 
Another definition of one coulomb: one coulomb is the amount of electric charge transferred by one ampere of current in one second of time. This definition is based on electric current that will be discussed later in this course.
_{Conservation law of electric charge: total amount of electric charge inside an electrically isolated system stays constant over time. This is one of the fundamental laws of physics.}

Coulomb's Law

If there are two point-like charges (or if the distance between them is very large compared to their dimensions), the electric force between them is defined by Coulomb’s law.
 According to the Coulomb’s law, the electric force between two point-like charges is directly proportional to the product of two charges and is inversely proportional to the square of distance between them; and it is also directed along a line joining these two charges.

 

In a coordinate system with origin 'o' , position vectors of two point-like charges 'q₁' , and 'q₂' , are 'r₁' , and 'r₂' respectively. 

 

The displacement vector from 'q₁'  to 'q₂'  can be written as,

and electric force acting on 'q₁'  by 'q₂', can be written as,

where

is the unit vector on vector 'r'.  

According to the Newton's third law, 'q₂' also exerts a force on 'q₁', given by,

These forces are action-reaction pair.

K in Coulomb’s law is proportionality constant, and its value depends on the medium; but its value in air is very close to its value in vacuum that is,

You can relate this value to the permittivity of vacuum (or air), using this equation,

where

is the permittivity of vacuum or air. When there are more than two point-like charges, the net electric force acting on one of them is the vector sum of electric forces exerted on it by the other objects. For instance, the net electric force acting on q₁ by q₂ , q₃ , ..., q_N , is given by,

 

 Electric Field

Each electrically charged object creates electric field in space with intensity 'E(r)' at any given point represented by a position vector 'r' in a coordinate system in space.

So at any given point the intensity of electric field depends on its position; and if there is a particle with electric charge q at this point, the electric force acting on it, is also dependent upon the position vector of this point, and it can be given by this equation:

Electric Field Lines

An electric field at a given point in space can be represented by a vector arrow with a length that corresponds to the magnitude of the electric field intensity at that point; the direction of the vector arrow indicates the direction of the electric field at that point.
In order to visualize the entire electric field around an electric charge, you can draw a set of lines with arrows on them, which represent the magnitude and direction of the electric field at every point of the space; these lines are called electric field lines. The direction of electric field at any point on a field line, is tangent to the line; and the strength of the field is proportional to the density of the field lines (the number of field lines passing through a unit area of a surface normal to the field lines at that point).
Figures below, show the electric field lines of some electric charge arrangements (note that the electric field vector at a point is tangent to the field line at that point):

                The electric field lines of a positive point charge +q are directed radially outward.

                 The electric field lines are directed radially inward towards negative point charge –q.

                          Electric field line pattern of two identical positive point charges.

 

                    Electric field line pattern of two identical negative point charges. 

                      Electric field lines of an electric dipole containing two opposite point charges.

 

 Spherically Symmetric Electric Fields

 Electric fields of point-like charges, and spherically symmetric distribution of charges, are spherically symmetric.

             Spherically symmetric fields created by point-like charges (or by spherically symmetric charges) in space.

In a coordinate system with origin at the center of a spherically symmetric electric field, electric field intensity at any given point, is given by,

 This equation comes from Coulomb's law, where  'r' is the magnitude of the position vector 'r', and 'r/r' is the unit vector on vector 'r'.
So the magnitude of electric field intensity at a given point in a spherically symmetric field is inversely proportional to the square of its distance from center of the field; direction of electric field at this point is the same as the direction of unit vector 'u' (or vector 'r') if electric charge 'q'  is positive, but if 'q' is negative, direction of electric field is opposite to the direction of unit vector 'u' (or vector 'r').
Electric charge 'q' in this equation, could be the electric charge of a point-like charge located at the center of the field (at the origin of the coordinate system) or it can be the total electric charge inside a sphere with radius 'r' if the electric field is created by a spherically symmetric distribution of charge.
If the distance of a spherically symmetric electric charge from the center of the field is larger than 'r', its contribution in the electric field created at point 'r' is zero, due to spherical symmetry.

 Electric Potential Energy

 Similar to a mass which have gravitational potential energy in a gravitational field, an electric charge also has electric potential energy in a electric field. Potential energy of a charge depends on its location in the field. When a charge changes its location in a field, its potential energy difference is equal to the work done on it in order to bring it from initial point to the final point. 
This work can be done by an applied force 'F_a', that is equal in magnitude but opposite to the direction of the force acting on the charge by the field.
So when an electric charge 'q' changes its location in the electric field 'E', its electric potential energy difference is given by,

  
Electric potential difference between two points, is also given by,

 that is electric potential energy difference of a unit charge.
In infinity potential energy of a charge goes to zero; so the electric potential energy of an electric charge 'q', at a given point 'r', in the electric field 'E', is equal to the energy needed to bring it from infinity to that point, i.e.,

and also the electric potential of that point, is,

So in addition to the field intensity that is a vector quantity, at any given point you can represent the electric field by the electric potential that is a scalar.

SI unit of electric potential is Volt. If the change in electric potential energy of one coulomb charge, is one joule when it moves from a point to the another point, the electric potential difference between these points is one volt.

In a spherically symmetric field, electric potential at any given point, can be written as,

 where 'Q' is either a point-like charge located at the center of the field, or a spherically symmetric charge inside a sphere with radius 'r' .

If 'Q' is a point-like charge, the equation above can be written as,

 

that is the electric potential at a distance 'r' from a point-like charge 'Q'.