Coulomb's Law & Its Applications

Table of Contents
1. Coulomb's Law
2. What is Relative Permittivity?
3. What is Dielectric Constant?
4. Coulomb's Law in Vector Form
5. Equilibrium and Stability under Coulomb Forces
6. Key Points on Coulomb's Law
7. Applications of Coulomb's Law
8. Problems on Coulomb's Law
9. Limitations of Coulomb's Law
10. Forces Between Multiple Charges (Superposition)
11. Solved Examples (Worked Answers)
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Coulomb's Law

Charles Augustin de Coulomb, in 1785 through his experiments, found that two point charges q₁ and q₂, kept at a distance r in a medium, exert an electrostatic force F on each other. The magnitude of this force is given by

This expression gives the magnitude of the electrostatic force experienced by q₁ due to q₂ and vice versa.

Coulomb's Law

In the expression above,

• F is the magnitude of the electrostatic force.
• q₁ and q₂ are the magnitudes of the two interacting point charges.
• K is the electrostatic constant which depends on the medium surrounding the charges. For vacuum, K = 1 / (4πε₀), where ε₀ (epsilon naught) is the permittivity of free space.

K = 1 / (4πε₀) = 9 × 10⁹ N·m²/C²
ε₀ = 8.854 × 10^-12 C²/N·m²

If the point charges are kept in a medium with permittivity ε, the electrostatic force between them becomes

The force acts along the line joining the two charges. It is repulsive if q₁ and q₂ have the same sign, and attractive if they have opposite signs. In words: like charges repel and unlike charges attract.

Permittivity of Free Space

Vacuum permittivity is another name for the permittivity of free space. Permittivity of open space is given as:
ε₀ ≈ 8.854187817620 × 10^-12F/m

Farad/meter is the SI unit for permittivity, often known as permittivity of open space. The dimensional symbol for the permittivity or permittivity of free space is [M^-1L^-3T⁴I²]

What is Relative Permittivity?

The permittivity of a substance relative to the permittivity of vacuum is referred to as relative permittivity. The Coulomb force between charged sites of a substance is described by a property of a material called permittivity. The electric field (between two charged sites) is reduced in relation to the vacuum as a result of this component.

Relative Permittivity of Free Space

The vacuum embodies permittivity at its lowest level. This is also known as the electric constant or the permittivity of free space. Denoted by ε₀ and has the value 8.85 x 10^-12 Farad/meter. Dielectrics exhibit the same resistance to the development of electric field lines. The relative permittivity of a dielectric, also known as a dielectric's permittivity, is defined as the ratio of the dielectric's absolute permittivity to the electric constant. It is described as an dimensionless quantity and is given as:
ε_r = ε/ε0

Where, ε₀ is the electric constant, ε_r is the relative permittivity and ε is the absolute permittivity of that material.

Relative Permittivity for Coulomb's Law

Using Coulomb's law, the magnitude of the electrostatic force between two point charges q₁ and q₂ separated by a distance r in free space can be calculated using relative permittivity (ε_r). By taking "the ratio of electrostatic force (F_a) between two point charges separated by a certain distance in air or vacuum to the electrostatic force (F_m) between the same two point charges separated by the same distance in a medium." It is written as:

What is Dielectric Constant?

Dielectric Constant Formula

• Polarization: Polarization can be considered as an event that occurs when positive and negative charges align within the dielectric but there is no overall increase in the dielectric's charge. A vector quantity P called polarization describes the degree of polarization of a dielectric.
• Electron susceptibility: When an electric field is created in an air dielectric substance. It polarizes electrically as a result. In the majority of materials, polarization and electric field are inversely correlated, that is,

P ∝ E
⇒ P = XeE
Where X_e is constant, a property of a substance known as electrical susceptibility.
X_e = P/E
Dielectric constant and susceptibility are related as:
D = ε_o(E+P) -----(1)
Also,
D = εE and P = X_eE
By changing these values, in equation (1),

It is mathematically expressed as
K = ε/ε₀
Where, K is the dielectric constant, ε is the permittivity of the substance and ε₀ is the permittivity of the free space.
It is a unitless, dimensionless quantity since it is the ratio of two like entities. The Greek letter kappa 'K' is used to represent the relative permittivity of a dielectric substance, which is also known as the dielectric constant.

Coulomb's Law in Vector Form

Coulomb's Law in Vector Form

To express Coulomb's law as a vector equation, consider two point charges q₁ at position r₁ and q₂ at position r₂. The vector from q₁ to q₂ is r = r₂ - r₁. The electrostatic force on q₂ due to q₁ is

Here F₁₂ denotes the force on charge 1 due to 2, and F₂₁ denotes the force on charge 2 due to 1. Coulomb's law applies for stationary (static) point charges. The forces obey Newton's third law:

For a charge placed in the presence of many other point charges, the total electrostatic force is the vector sum of forces due to individual charges (superposition):

What is 1 Coulomb of Charge?

A coulomb is that amount of charge which, if placed as two equal point charges one metre apart in vacuum, repels each other with a force of 9 × 10⁹ N.

Note: Coulomb force is valid only for static charges and when particles can be treated as point charges (or spherically symmetric charge distributions).

Equilibrium and Stability under Coulomb Forces

Consider equilibrium conditions for a small charge placed under the influence of two fixed charges located at A and B. The character of equilibrium depends on the direction of small displacements:

• If the charge is displaced slightly along the line joining A and B (axial displacement), one of the forces increases while the other decreases; the net force may drive the charge further away from the equilibrium point. Thus axial displacement tends to produce unstable equilibrium.
• If the charge is displaced slightly perpendicular to AB, the forces due to A and B tend to restore the charge to its original position. Thus perpendicular displacement tends to produce stable equilibrium.

Key Points on Coulomb's Law

• If the force between two charges in two different media is the same for different separations, then Kr²= constant.
  
• Therefore Kr² = constant or K₁r₁² = K₂r₂².
• If the force between two charges separated by distance r₀ in vacuum equals the force between the same charges separated by distance r in a medium, then Kr² = r₀².
• Two identical conductors carrying charges q₁ and q₂ are put into contact and then separated; each will have charge (q₁ + q₂)/2. If the charges are q₁ and -q₂, then after contact each will have (q₁ - q₂)/2.
• Two spherical conductors of radii r₁ and r₂, carrying charges q₁ and q₂, are put into contact and then separated. After contact, the charges become: q₁' = [r₁/(r₁ + r₂)](q₁ + q₂) and q₂' = [r₂/(r₁ + r₂)](q₁ + q₂).
• If the force between two identical conductors with charges q₁ and q₂separated by distance d is F, and the conductors are first put to contact and then separated by the same distance, the new force is given by
  
• If charges are q₁ and -q₂, then F = F(q₁ + q₂)² / (4q₁q₂).
• Between two electrons separated by a given distance: Electrical force / Gravitational force ≈ 10⁴².
• Between two protons separated by a given distance: Electrical force / Gravitational force ≈ 10³⁶.
• Between a proton and an electron separated by a given distance: Electrical force / Gravitational force ≈ 10³⁹.
• The relationship between the velocity of light c, the permeability μ₀ and the permittivity ε₀ of free space is c = 1 / √(μ₀ ε₀).
• If two identical charged balls of mass m are hung by threads of length l from the same point and carry equal charges q, then:
  
  The distance between the balls is given by the figure/equation above.
  
  The tension in a thread is given by the expression shown above.
  
  If the system is kept in space so that the threads are in a straight line, the angle between threads is 180° and the tension is given by the expression in the figure. A charge Q divided into q and (Q - q) produces maximum electrostatic force between parts when
  

Applications of Coulomb's Law

• To calculate the force between two point charges and the distance at which a specified force acts.
• To calculate the electric field produced by a point charge using E = F / Q_T where Q_T is a test charge.
• To calculate the force on a charge due to several other charges by the principle of superposition.

In the figure, E is the electric field (strength), F is the electrostatic force on the test charge, and Q_T is the test charge in coulombs.

Problems on Coulomb's Law

Problem 1: Charges of magnitude 100 microcoulomb each are located in vacuum at the corners A, B and C of an equilateral triangle measuring 4 meters on each side. If the charge at A and C are positive and the charge B negative, what is the magnitude and direction of the total force on the charge at C?
Sol. 

The situation is shown in the figure. Consider the forces acting on C due to A and B.

Now, from Coulomb's law, the force of repulsion on C due to A (F_CA) in direction AC is given by

The force of attraction on C due to B (F_CB) in direction CB is given by

Thus the two forces are equal in magnitude. The angle between them is 120°. The resultant force F is given by

This resultant force is parallel to AB.

Problem 2: The negative point charges of unit magnitude and a positive point charge q are placed along the straight line. At what position and for what value of q will the system be in equilibrium? Check whether it is stable, unstable or neutral equilibrium.
Sol. 

The two negative charges A and B of unit magnitude are shown in the figure. Let the positive charge q be at distances r_A from A and r_B from B.

From Coulomb's law, force on q due to A is

Force on q due to B is

These two forces acting on q are opposite and collinear. For equilibrium of q, the two forces must be equal in magnitude:

|F_qA| = |F_qB|

Hence r_A = r_B. So q must be equidistant from A and B, i.e., at the midpoint of AB.

Now for the equilibrium of the entire system, A and B must also be in equilibrium. For equilibrium of A:

Force on A by q is

Force on A by B is

The two forces are opposite and collinear. For equilibrium they must be equal and opposite. Hence

Thus q = 1/4 in magnitude of either unit charge. Similarly, for equilibrium of B, q must be 1/4 of the magnitude of either charge.

Problem 3: A positive charge of 6 × 10^-6 C is 0.040 m from the second positive charge of 4 × 10^-6 C. Calculate the force between the charges.
Given
q₁ = 6 × 10^-6 C
q₂ = 4 × 10^-6 C
r = 0.040 m
Sol. 

F_e = 134.85 N

Problem 4: Two-point charges, q₁ = +9 μC and q₂ = 4 μC, are separated by a distance r = 12 cm. What is the magnitude of the electric force?
given
k = 8.988 × 10⁹ N·m²·C^-2
q₁ = 9 × 10^-6 C
q₂ = 4 × 10^-6 C
Sol:

F_e = 22.475 N

Limitations of Coulomb's Law

Coulomb's law is derived under specific assumptions and therefore has limitations:

• It applies only to static (stationary) charges.
• It is straightforward for charges of regular, smooth shapes; complications arise for irregular charge distributions where the point-charge approximation fails.
• It is valid when the medium between charges can be treated as a continuous homogeneous medium with well-defined permittivity; microscopic solvent structure or molecular-scale separations can invalidate the simple form.

Forces Between Multiple Charges (Superposition)

Charge is an intrinsic property of matter. An atom becomes charged when the numbers of electrons and protons are not equal. Common charging methods include rubbing (triboelectric effect); for example, rubbing a plastic comb with hair transfers electrons to or from the comb, enabling it to attract small pieces of paper.

Calculating force using superposition

The force between any two charges is given by Coulomb's law. When several charges are present, the net force on a chosen charge is the vector sum of forces due to each of the other charges (principle of superposition).

For three point charges Q_A, Q_B and Q_C at position vectors r₁, r₂ and r₃, the net force on the charge at r₁ is

This can be written as

Applying this to the specific three-charge example gives

• The force acting on a charge is directly proportional to the magnitude of the other charges and inversely proportional to the square of the distance between them.
• The net force on a point charge due to several charges is the vector sum of the individual forces.

Solved Examples (Worked Answers)

Q.1. Two charges 1 C and -3 C are kept at a distance of 3 m. Find the force of attraction between them.
Ans. 

We have q₁ = 1 C, q₂ = -3 C and r = 3 m.

Using Coulomb's law:

F = k q₁ q₂ / r²

Substituting values:

F = 9 × 10⁹ × (1) × (3) / 3²

F = 3 × 10⁹ N

Q.2. The electron and proton of a hydrogen atom are separated (on the average) by a distance of approximately 5.3 × 10^-11 m. Find the magnitudes of the electric force and the gravitational force between the two particles.
Ans.

Treat the electron and proton as point particles separated by r = 5.3 × 10^-11 m in vacuum.

Given:

• Mass of electron m_e = 9.1 × 10^-31 kg
• Mass of proton m_p = 1.67 × 10^-27 kg
• Charge of electron q_e = -1.6 × 10^-19 C
• Charge of proton q_p = +1.6 × 10^-19 C

By Universal Law of Gravitation:

By Coulomb's Law:

The electric force between the electron and proton is approximately 8.2 × 10^-8 N.

Note: The electrostatic force between the proton and electron is about 10³⁹ times larger than the gravitational force between them. Hence gravity is negligible for charged microscopic particles compared with electrostatic forces.