Class 12 Physics Chapter 1 – Electric Charges and Fields

## Forces between Two Point Charges – Coulomb’s Law, Coulomb’s Law in Vector Form

**[Watch Video to Understand in Detail and in Malayalam Language]**

### Coulomb’s Law

Coulomb’s law
states that the force between two point charges* varied inversely as the
square of the distance between the charges and was directly proportional to the
product of the magnitude of the two charges and acted along the line joining
the two charges*.

Thus, if two point charges *q*_{1}, *q*_{2} are separated by a distance *r *in vacuum, the magnitude of the force (F) is

and

In free space or vacuum the value of constant *k*

is called the *permittivity of free space. *The value of in SI units is

So that Coulomb’s law is written as

The Coulomb’s law and the value of gives us an idea about SI unit of charge coulomb. Let q_{1} = q_{2} = 1 C and r = 1 m, then force

That is, 1 C is the charge that when placed at a distance of 1 m from
another charge of the same magnitude in vacuum experiences an electric force of
repulsion of magnitude 9 X 10^{9} N. 1 C is too big unit to be used. In
practice, in electrostatics, one uses smaller units like 1 mC or 1 µC.

### Coulomb’s law in Vector Notation

Since force is a
vector, we have to write Coulomb’s law in the vector notation [Note: here we
are using bold letters to represent vectors]. Let the position vectors of
charges *q*_{1} and *q*_{2} be **r**_{1}
and **r**_{2} respectively. The resultant vector leading from q_{1}
to q_{2} is denoted by

**r**_{21} = **r**_{2} – **r**_{1}

And the vector
leading from q_{2} to q_{1} is denoted by **r**_{12}:

**r**_{12} = **r**_{1} – **r**_{2
}= – **r**_{21}

The unit vectors along this can be represented as:

Also force on *q*_{1}
due to *q*_{2} by **F**_{12} and force on *q*_{2}
due to *q*_{1} by **F**_{21}.

Coulomb’s force law between two point charges *q*_{1} and *q*_{2} located at **r**_{1} and **r**_{2} in vacuum or free space is then expressed as

See figure for geometric representation.

The above diagram is for q_{1 }x q_{2} > 0, the force is repulsive in nature. If q_{1 }x q_{2} < 0, the force is attractive in nature and figure below shows that.

Some points related to Coulomb’s law in vector notation are as follows:

It is valid for any sign of *q*_{1} and *q*_{2} whether positive or negative. If *q*_{1} and *q*_{2} are of the same sign (either both positive or both negative), **F**_{21 }is along , which denotes repulsion, as it should be for like charges. If *q*_{1 }and *q*_{2 }are of opposite signs, **F**_{21} is along , which denotes attraction, as expected for unlike charges. That is equation is true for both cases correctly.

It also proves that Coulomb’s law agrees with Newton’s third law.

The force **F**_{12} on charge *q*_{1} due to charge *q*_{2}, is obtained by simply interchanging 1 and 2 in the above equation, i.e.,