Coulomb’s Law – Derivation, Limitations & Vector Form

by Snir Mac
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This experimental law was presented by a French military engineer and physicist Charles Agustin De Coulomb (1736-1806).

Statement

It states that “The magnitude of force between any two point charges either attractive or repulsive, is directly proportional to the product of charges and inversely proportional to the square of the distance between their centres”.

Derivation

Consider two point spherical charges of magnitude q1 and q2 with distance of separation between them being r. Force F is exerted between these two charges where is a unit vector indicating the direction of force.

According to Coulomb’s Electrostatics Law, force F experienced is:

► Directly proportional to the product of two point charges
F ∝ q1q2
► Inversely proportional to the square of the separation between the charges
F ∝ 1 / r2
► Combining these two relations, we get :
F ∝ q1q2 / r2
F = kq1q2 r̂ / r2

is a unit vector which indicates the direction of force F acting between the two charges q1 and q2
k is a proportionality constant equal to 1 / 4πεo (= 9 x 109 Nm2 C-2).
εo is known as permittivity of free space or vacuum equal to 8.85 x 10-12 C2 Nm-2

Vector Form Of Coulomb’s Law

Vector form of two oppositely charged bodies

In the given figure above, 21 is a unit vector pointing towards q1 which shows the direction of force F2 on charge q1 due to charge q2 .

Similarly, 12 is a unit vector pointing towards q2 which shows the direction of force F1 on charge q2 due to charge q1.

Thus, the force experienced by charge q1 due to charge q2 is F2. According to Coulomb’s Law, we have:

F2 = kq1q212 / r2 —> (1)

F1 = – kq1q221 / r2 —> (2)

From the figure, it is clearly depicted that both the unit vectors 12 and 21 are oppositely directed. Hence, their forces F2 and F1 will be also oppositely directed i.e negative in sign.

Comparing equations (1) and (2), we get:

F1 = – F2

This shows us that Coulomb’s Law of Electrostatics conforms with Newton’s third law of motion.

Coulomb’s Law in Material Media (Dielectric)

When the medium between the two charges is other than air or vacuum then the force decreases by an amount εr .

ε is known as the permittivity of material media. For convenience, we replace ε by product of ε0 (Permittivity of free space) and εr (Relative permittivity or Dielectric constant)

Permittivity is the measurement of resistance to the expansion of electric field lines in a medium.

According to electrostatic force of attraction between two charged bodies relatively at rest, its mathematical equation can be deduced as follows:

The medium between any two charges is vacuum or air then electrostatic force of attraction is given by:

Fvac = q1q2 / 4πεo r2 —> (3)

And if the medium between any two charges is replaced by other medium i.e. Dielectric then electrostatic force of attraction is given by:

Fmed = q1q2 / 4πεoεr r2 —> (4)

From the above equation (4), it is clear that F decreases by an amount εr (Dielectric constant of the medium) when dielectric is placed between the two charges other than vacuum or air.

Fmed = (1/εr) q1q2 / 4πεo r2 —> (5)

The above equation can also be written as:

Fmed = (1/εr) Fvac —> (6)

Limitations of Coulomb’s Law

  • Coulomb’s Law is only applicable to point charges.
  • Coulomb’s Law is only applicable to spherical charges.
  • Coulomb’s Law is only applicable for charges at relative rest.

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