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Coulomb Law example problems with solutions

What does Coulomb’s law mean?

“Coulomb’s law states that  force of attraction or repulsion between two electric charges is directly proportional to their magnitudes and inversely proportional  to the square of distance between them”.It is mathematically explained as:

Charles Agustin Coulomb (1736-1806) measured electrical attractions and repulsions  quantitatively and deduced the law that governs them.His apparatus,shown in fig.

coulomb's law aparatus

This figure resembles the figure given below:

charge production

If a and b are charged,the electric force on a tends to twist the suspension fiber.Coulomb cancelled out this twisting effect by turning the suspension head through the angle θ needed to keep the two charges at a particular separation.The angle θ is then a relative measure of the electric force acting on charge a.The device of above figure is a torsion balance,a similar arrangement was used latter by Cavendish to measure gravitational attractions.

coulomb’s law equation:

mathematical form of coulomb's law

This relation is called coulomb’s law.Here F is called the magnitude of the mutual force that acts on each of the two charges a and b ,qand q2 are relative measures of the charges on spheres a and b,and r is the distance between their centers.The force on each charge due to the other acts along the line connecting the  charges.The two forces point in opposite directions but have equal magnitudes,even though the charges may be different.The two forces point in opposite directions but have equal magnitudes ,even though the charges may be different.

What is the value of k in coulomb’s law?

To turn the above proportionality into an equation,let us introduce a constant of proportionality,which we represent for now as k .Its value depends upon the system of units and medium between the charges.For free space and in system international ‘k’ is expressed as:

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coulomb's constant

Where ε is the permittivity of free space and its value is 8.85 × 10-12C-2N-1m-2.

we thus obtain,for the force between the charges.Coulomb’s law generally holds only for charged objects whose sizes are much smaller than the distance between them.We often say that it holds only for point charges.

Our belief in coulomb’s law does not rest quantitatively on coulomb’s experiments.Torsion balance measurements are difficult to make to an accuracy of better than a few percent.

Coulomb’s law resembles Newton,s inverse square law of gravitation,which was already more than 100 years old at the time of Coulomb’s experiments.Both are inverse square laws,and the charge q plays the same role in Coulomb’s law that the mass m plays in Newton’s law of gravitation.One difference between the two laws is that gravitational forces,as far as we know,are always attractive,while electrostatic forces can be repulsive or attractive,depending on whether the two charges have the same or opposite signs.

There is another important difference between the two laws.In using the law of gravitation,we were able to define mass from Newton’s second law,and then by applying the law of gravitation to know masses we could determine the constant G.in using Coulomb’s law ,we take the reverse approach;we define the constant k to have a particular value,and we then use Coulomb’s law to determine the basic unit of electric charge as the quantity of charge that produces a standard unit of force.

For example,consider the force between two equal charges of magnitude q.We could adjust q until the force has a particular value,say 1 N for separation of r=1m,and define the resulting q as the basic unit of charge.It is,however,more precise to measure the magnetic force between two two wires carrying equal currents,and therefor the fundamental SI electrical unit is the unit of current,from which the unit of charge is derived.The operational procedure for defining the SI unit of current ,which is called the ampere.

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Vector form of Coulomb’s law:

Coulomb’s force is a mutual force ,it means that if charge ‘q1‘ exerts a force on charge ‘q2‘ then q2 also exerts an equal and opposite force on q1.

vector form of coulomb law

If charge qexerts an electrostatic force “F12 charge q2 and q2 exerts electrical force F21 on charge q1.

If r12 represents the direction of force from charge qto charge qand r21 is the unit vector which represents the direction of force from charge  qto q1,then:

derivation of vector form of coulomb's law

According to newton’s 3rd law:


This expression shows that coulomb force is a mutual force.

Significance of vector form of coulomb’s law:

Vector form of coulomb’s law has the critical importance,when there is an assembly of point charges.In this case ,the resultant force on any one of the charges is the vector sum of the forces due to each of the other forces.This is called principle of superposition.

Comparison of Coulomb’s law and Newton’s law of gravitation:

The electrical force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them:It is expressed

coulomb's law

The gravitational force between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between them:

newton's gravitation law

Similarities among the electrical and Gravitation force:

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Differences among the electrical and gravitational force:

  • Electrical force is might be attractive as well as repulsive,while the gravitational force is only attractive.
  • Electrostatic force is medium dependent and can be shielded while the gravitational force lacks this property.
  • The value of gravitational constant is very small while the electrical constant is very large.It is because of the fact that gravitational force is very weak as compared to electrical force.
  • In using the law of gravitation,we define mass ‘m’ from F=ma ,then determine ‘G’ by applying gravitational law to known masses.While in case of coulomb’s law,we define ‘k’ for particular value and then determine ‘q’ by applying coulomb’s law.

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