Coulomb’s Law: Formula, Vector Form & Limitations

Coulomb’s Law gives the relation between the force of attraction or repulsion between two charged bodies and the distance between the two bodies. As per Coulomb’s Law:

“The force of attraction or repulsion between two charged bodies is directly proportional to the product of its charges and inversely proportional to the square of their distance.”

Coulomb’s law can be simply defined as a law explaining the amount of force between two stationary, and electrically charged particles (also termed as electrostatic force). Coulomb’s law formula can be represented by:

F_e = \(\frac{1}{4\pi \epsilon_0 K} . \frac{q_1q_2}{d^2}\) = \(\frac{1}{4\pi \epsilon_0 \varepsilon_r} . \frac{q_1 q_2}{d^2}\)

Here,

• ε = Permittivity of the medium,
• K or ε_r = Relative permittivity or specific inductive capacity 
• ε₀ = Permittivity of free space, also known as absolute permittivity

Note: K or ε_r is also termed the “dielectric constant” of the medium where the two charges are located.

It is one of the most important laws given by a French Physicist, Charles Augustin de Coulomb in 1784. Coulomb measured the force between two point charges and coined a relationship between two charged bodies. The equation is known as Coulomb’s Law or Coulomb’s inverse-square law.

Also Read: Electric Charges and Fields 

Key Terms: Coulomb’s Law, Coulomb’s Law Formula, Vector Quantity, Principle of Superposition, Electric Field, Electric Charge

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What is Coulomb’s Law?

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Coulomb’s Law gives the magnitude of the force acting between two electrically charged bodies separated by a certain distance. Coulomb’s Law can be expressed as:

Let’s consider two charged bodies q₁ and q₂. The distance between these two charges is “r”, and the force of attraction or repulsion between the bodies is “F_e”.

Coulomb's Law

Coulomb’s First Law

Coulomb’s first law claims that the like-charged bodies repel one another, while the unlike-charged bodies attract one another.

Coulomb’s Second Law

Coulomb’s second law expresses that “the force of attraction or repulsion between two electrically charged bodies is in direct proportion to the magnitude of their charge and inversely proportional to the square of the distance between the bodies.” Therefore, as per Coulomb’s second law,

F \(\propto\) Q₁ Q₂ and F \(\propto\)\(\frac{1}{d^2}\)

Thus,

⇒ F \(\propto\) \(\frac{Q_1 Q_2}{d^2}\)

⇒ F = \(k\frac{Q_1 Q_2}{d^2}\)

Illustration of Coulomb's law

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Coulomb's Law Formula

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Coulomb’s Formula (in short) can be given by:

F_e ∝ \(\frac{Q_1 Q_2}{d^2}\)

⇒ F_e = \(k\frac{Q_1 Q_2}{d^2}\)

Where,

• “k” is a constant known as Coulomb's constant or Electric force constant, and its value is equal to \(\frac{1}{4\pi \varepsilon_0}\)

Thus, the formula of Coulomb's Law is,

Here,

• ε₀ is the permittivity of free space or absolute permittivity
• ε_r or K is the relative permittivity of the medium, also known as the dielectric constant of the medium
• ε is the permittivity of the medium = ε_rε_o

The value of absolute permittivity is ε₀ = 8.854 × 10^-12 C² N^-1 m^-2

Since the relative permittivity of air ε_r = 1, therefore if the two charges are located in the air, then the electrostatic force between them is given by

\(F=\frac{1}{4\pi \epsilon_o}\frac {q_1q_2}{r^2}=k\frac {q_1q_2}{r^2}\)

The value of Coulomb's constant is k = 9 × 10⁹ Nm² / C²

Coulomb's Law Formula

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Coulomb’s Law in Vector Form

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Physical quantities are either in scalar form or vector form. As force is a vector quantity, Coulomb’s Law can be stated in vector form as follows: 

Coulomb's Law in Vector Form

Mathematically the vector form of Coulomb’s law can be given as,

\(\begin{array}{l}\rightarrow{{\vec{F}}_{12}}=\frac{1}{4\pi {{\in }_{0}}}\frac{{{q}_{1}}{{q}_{2}}}{r_{12}^{2}}{{\hat{r}}_{12}}; \;\;{{\vec{F}}_{12}}=-{{\vec{F}}_{21}}\end{array}\).

\(\overrightarrow{F}_{12}\) = – \(\overrightarrow{F}_{21}\) (For repulsion)

\(\overrightarrow{F}_{12}\) = + \(\overrightarrow{F}_{21}\) (For attraction)

Here,

• \(\overrightarrow{F}_{12}\) = The force exerted by q₁ on q₂
• \(\overrightarrow{F}_{21}\) = The force exerted by q₂ on q₁

Coulomb’s law is known to hold for stationary charges that are point sized. Thus, the law follows Newton’s third law.

The force on a charged particle which is due to point charges is the resultant of forces due to individual point charges. Thus,

⇒ \(\vec{F}={{\vec{F}}_{1}}+{{\vec{F}}_{2}}+{{\vec{F}}_{3}}+……\)

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One Coulomb Charge

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1 coulomb is equal to a point charge placed at a distance of 1 meter from another point charge of the same magnitude in a vacuum with a repulsion force of magnitude 9 × 10⁹ N.

The value of ε₀ = 8.86 × 10^-12 F/m or 8.86 × 10^-12 C² N/ m² .

Coulomb’s Law (Conditions for Stability)

If q displaces slightly towards A, F_A increases in magnitude and F_B decreases in magnitude.

• Hence, the net force on q can now be found towards A, which is why it will not return to its original position.
• Thus, for axial displacement, the equilibrium is going to be unstable.
• Assuming that q is perpendicularly displaced to AB, the force F_A and F_B will obtain the charge to its original position.
• Thus, for perpendicular displacement, the equilibrium is going to be stable.

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Principle of Superposition

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Coulomb’s law satisfies the principle of superposition, which means that the force between two particles is not affected by the presence of other charges.

• This helps in finding the net force exerted on a charged particle by other charged particles.
• The force on a charged particle q₀ due to charges q₁, q₂, and q₃, is the net force due to individual point charges.
• Hence, \(\overrightarrow{F}_{0} = \overrightarrow{F}_{01} + \overrightarrow{F}_{02} + \overrightarrow{F}_{03}\)

Relative Permittivity of a Material

The relative permittivity of a material can be described as:

\({{\varepsilon }_{r}}=K=\frac{force\;between\;two\;charge\;in\;air}{force\;between\;same\;charge\;in\;the\;medium\;at\;the\;same\;distance}\)

Thus, \({{\varepsilon }_{r}}=\frac{F_a}{F_m}\)

• In case of air, K = 1
• In case of metals, K = infinity

The force between two charges is based upon the nature of the medium that is intervening, while gravitational force is independent of the medium which is intervening.

Thus, for air or vacuum, \(F=\text{ }\frac{\text{1}}{\text{4}\pi {{\varepsilon }_{\text{o}}}}.\frac{{{\text{q}}_{\text{1}}}{{q}_{2}}}{{{d}^{2}}}\)

Hence, the value of 1/4πε₀ is = 9 × 10⁹ Nm²/C².

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Limitations of Coulomb’s Law

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Coulomb’s Law cannot be used freely like other laws as it is derived under certain assumptions. Here are some limitations of Coulomb’s law:

• The formula can only be applied on static charges (which are in the rest position).
• The application of Coulomb’s law when the charges are in irregular shape is quite complex, as it is difficult to determine the distance between the charges in this case.
• For regular shapes, the law can be used easily.
• Coulomb’s formula is only valid when the solvent molecules between the particles are larger than both the charged bodies.
• The law cannot be implemented directly to find the charge on big planets.

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Coulomb’s Law Application

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Coulomb’s law can be implemented in the following cases.

• To calculate the distance or force between two point charges
• To calculate the force on one point due to several other point charges.
• To calculate the strength of the Electric Field. The formula for calculating the electric field using Coulomb’s Law is as follows:

\(E= \frac{F}{Q}\times \frac{N}{C}\)

Where,

• E = Strength of Electric Field
• F = Electrostatic Force
• Q = Test charge in Coulomb’s

Coulomb's law application

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Key Points of Coulomb’s Law

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Some important points of Coulomb’s Law are:

• Assuming that the force between two charges found in two different media is the same for different separations. Thus, \(\begin{array}{l}F=\frac{1}{K}\frac{1}{4\pi {{\in }_{0}}}\frac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}} = constant\end{array}\)
• Kr² = constant or K₁r₁² = K₂r₂² 
• Assuming that the force between two charges that are separated from one another by a distance ‘r₀’ in a vacuum is equal to the force between the same charges now separated by a distance ‘r’ in a medium, then Coulomb’s Law will be: Kr² = r₀²
• Two identical conductors with q₁ and q₂ charges are connected and further separated. Thus, each will have a charge equivalent to (q₁ + q₂)/2. In case the charges are q₁ and – q₂, each will have a charge equivalent to (q₁ – q₂)/2.
• Two spherical conductors with q₁ and q₂ charges and radii r₁ and r₂ are first brought in contact and further separated. Thus, the charges of the conductors post-contact is; q₁ = [r₁/(r₁ + r₂)] (q₁ + q₂) and q₂ = [r₂/(r₁ + r₂)] (q₁ + q₂)
• The force of attraction or repulsion between two identical conductors with q₁ and q₂ charges separated by a distance d is F. After they are brought into contact and then separated by the same distance, then the new force between them becomes: \(F = \frac{F{{\left( {{q}_{1}}+{{q}_{2}} \right)}^{2}}}{4{{q}_{1}}{{q}_{2}}}\)

• If charges are q₁ and -q_2, then F = F(q₁ + q₂)² / 4q₁q₂

• In the case of two electrons being separated by a certain distance, then the ratio of electrical force to the gravitational force is 10⁴².

• In the case of two protons being separated by a certain distance, then the ratio of electrical force to gravitational force is 10³⁶.

• In the case of an electron and a proton being separated by a certain distance, then the ratio of electrical force to the gravitational force is 10³⁹.

• The relationship that can be established between the velocity of light, the permeability of free space, and the permittivity of free space can be represented by, c = 1 / √ (μ_oε_o )

•  By applying Coulomb’s law to two identical balls of mass m that are hung by a silk thread of length ‘l’ from the same hook with charges q, then:

1. Distance Between balls = \(\frac{q^{2}2l }{4\pi \epsilon _{o}mg}]^{\frac{1}{3}}\)
2. Tension in Thread = \(\sqrt{f^2+(mg)^2}\)
3. The total system is kept in space, with an angle between the thread as 180°. Thus, the tension in a thread: \(T=\frac{1}{4\pi {{\in }_{0}}}\frac{{{q}^{2}}}{4{{\ell }^{2}}}\)
4. A charge Q when divided between q and (Q – q), the electrostatic force between them becomes maximum: \(\frac{q}{Q}=\frac{1}{2}\;\ (or) \;\;\;\frac{q}{\left( Q-q \right)}=1\)

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How to Solve Coulomb’s Law?

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Below are the steps to follow while solving problems on Coulomb’s Law:

• Determine whether the force due to the charge is attractive or repulsive and represent it by drawing a vector pointing toward or away from the given charge, respectively.
• Determine the magnitude of force using Coulomb’s Law (without considering the signs of charges).
• Solve the forces with the given coordinate axis and express them in vector form using \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) unit vector notation.
• Apply the principle of superposition to find the net force on the charge.

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Things to Remember

• When the force between two charges in different media is the same for different separations, F = \(\frac{1}{4\pi \varepsilon_0 \varepsilon_r} . \frac{q_1q_2}{r^2}\) = constant
• If the force between two charges at a distance “r₀” in a vacuum is the same as the force between the same charges at a distance “r” in a medium, then Coulomb’s law is given as Kr² = Kr₀²
• The charge produced by two similar charges is equal to (q₁ + q₂)/2.
• Alternatively, when the charges are q₁ and - q₂ then each of them will possess a charge equal to (q₁ - q₂)/2.
• For charges q₁ and - q₂, F =\(\frac{F(q_1 + q_2)^2}{4q_1q_2}\)
• The relationship between the velocity of light, the permeability of vacuum, and the permittivity of vacuum is denoted as c = 1/\(\sqrt{\mu_0 \varepsilon_0}\).

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Previous Year Questions 

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