Coulomb’s Law: Statement, Vector form, Nature of force and Limitation

coulomb’s law

If the number of charges is present in the media, the force on any charge due to other charges can be calculated by the coulomb’s law. Then total force obtains by adding all forces. This force is known as electrostatic force. The force is a vector quantity. It means force has magnitude as well as direction.

The electrostatic force was first given by a French physicist Charles-Augustin de Coulomb in 1785. He derived an equation to find the magnitude of repulsion or attraction force between two charged bodies. This relation is known as the coulomb’s law.

Coulomb’s law:

“Between two point charges, there is a force of attraction and repulsion depending upon the nature of the charge.”

There is a force of repulsion if the charges are like charges and for unlike charges, there is a force of attraction.

Statements:

a) The magnitude of the force is proportional to the product of the magnitude of charges.

(1)   \begin{equation*}   F \propto Q1;  \quad   F \propto Q2; \quad   F \propto Q1Q2; \end{equation*}

b) The magnitude of the force is inversely proportional to square of the distance between the charges.

(2)   \begin{equation*}   F \propto \frac{1}{R^2} \end{equation*}

c) The magnitude of the force is depending on the medium between two charges. Hence, It is proportional to the permittivity of the medium (ϵr). 

From eq (1), and (2)

(3)   \begin{equation*}  F \propto \frac{Q1Q2}{R^2} \end{equation*}

Hence, if we consider two-point charges Q1 and Q2 separated by R meter distance. The medium between two-point charge is free space. According to the coulomb’s law, the equation of force is given as below equation.

    \[  F = k \frac{Q1Q2}{R^2} \]

The constant of proportionality is given by,

    \[  \quad k=\frac{1}{4 \pi \epsilon_0 \epsilon_r} \]

    \[  \quad \epsilon = \epsilon_0 \epsilon_r \]

    \[  \epsilon_0\; =& Permittivity \; of \; space  \]

    \[  \epsilon_r\; =& Permittivity \; of \; medium \]

The proportionality constant is known as the electrostatic constant.

The vector form of coulomb’s law:

Force is a vector quantity. Force acts along the line joining the two charges. Therefore, above equation multiplied by the unit vector along the line. Hence, the vector form of coulomb’s law expressed as below equations.

Force on Q2 is

(4)   \begin{equation*}  \overline{F_2} = \frac{Q1Q2}{4 \, \pi \, \epsilon \, R^2_1_2} \overline{a_1_2} \end{equation*}

Force on Q1 is

(5)   \begin{equation*}  \overline{F_1} = \frac{Q1Q2}{4 \, \pi \, \epsilon \, R^2_2_1} \overline{a_2_1} \end{equation*}

The unit of force is newton (N).

Nature of force:

When the charges are like charges there is a force of repulsion. Two-point charges Q1 and Q2 are placed at distance R. The unit of charges is coulomb. For charges, +VE and –VE sign shows the polarity of charges. If both charges have positive polarity (+Q1 and +Q2) or both charges have negative polarity (-Q2 and -Q2), in this case, both charges experience the repulsive electrostatic force. Because of this force, both charges try to move far from each other.

coulomb’s law
coulomb’s law:

When the charges are unlike charges there is a force of extraction. Two-point charges Q1 and Q2 are placed at distance R. If both Q1 has a positive polarity and Q2 has negative polarity (+Q1 and -Q2) or Q1 has negative polarity and Q2 has positive polarity (-Q1 and -Q2), in this case, both charges experience the extractive force. Because of this force, both charges try to move near to each other.

Limitation of coulomb’s law:

  1. In coulomb’s law, we assume that all charges are a sphere in shape. But, if charges are in arbitrary shape, it is difficult to apply.
  2. Coulomb’s law is valid if the point charges are at rest.
  3. If the average number of solvent molecules between the two charge particles must be large, in this condition only coulomb’s law is applicable.

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