Coulomb’s Law - Coulomb's Law in Vector Form

Just as gravity pulls two masses toward each other along the line joining them, the electrostatic force between two charges also acts along the straight line joining the charges. Coulomb's Law in vector form expresses not just the magnitude of this force (as the scalar form does) but also its direction — telling us whether the charges attract or repel, and along which line the force acts.

This form is essential because forces are vector quantities, and any physical situation involving multiple charges requires vector addition of individual forces (superposition principle).

Statement

The electrostatic force acting between two stationary point charges is given by a vector quantity whose magnitude obeys Coulomb’s law and whose direction is along the line joining the two charges. The force on each charge is equal in magnitude and opposite in direction.

Explanation / Mathematical Form

Let two point charges q_1​ and q₂ be located at position vectors \[\vec {r_1}\]​ and \[\vec {r_2}\]​ respectively.

The force on charge q₁ due to charge q₂​ is:

\[\vec F_{12}\] = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{r}_{12}\]​

Similarly, the force on q₂​ due to q₁ is:

\[\vec F_{21}\] = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}^2}\hat{r}_{21}\]​

where
\[\hat r _{12}\]​ and \[\hat r_{21}\] are unit vectors along the line joining the charges and

​\[\hat r_{21}\] = -\[\hat r _{12}\]​

Hence,

\[\vec F_{21}\] = −\[\vec F_{12}\] ​

This relation is valid for both like and unlike charges, representing repulsion or attraction respectively.

Conclusion

The vector form of Coulomb’s law shows that:

• Electrostatic force is a central force acting along the line joining the charges.
• Forces between two charges are equal and opposite, satisfying Newton’s third law.
• The direction of force is clearly specified, unlike the scalar form.

Important: In calculations, always substitute charges with their sign (+ or −); the formula automatically gives the correct direction.

• When charges are placed in a dielectric medium (instead of vacuum/air), the force reduces due to the dielectric constant K:
  \[F_m=\frac{F_0}{K}\]
  where F₀​ is the force in vacuum/air and F_m​ is the force in the medium.
• The equivalent air distance relation (distance in air giving the same force as distance r in medium):
  \[r^{\prime}=r\sqrt{K}\]

Comparing Electrostatic and Gravitational Forces Between Protons

Step 1: Calculate the electrostatic force (F_e)

• Substitute the charge of a proton (1.6 × 10^-19 C) and the distance (10^-15 m) into Coulomb's law formula.
• Solving this yields an electrostatic force of 2.3 × 10² N.

Step 2: Calculate the gravitational force (F_g)

• Substitute the gravitational constant (G), the mass of a proton (1.67 × 10^-27 kg), and the distance into Newton's law of gravitation.
• Solving this yields a gravitational force of 1.86 × 10^-34 N.

Step 3: Compare both forces

• Divide the electrostatic force by the gravitational force to get a ratio of 1.23 × 10³⁶, showing the electrostatic force is vastly stronger.

Dielectric Equivalence in Air

Step 1: Write the force equation in a dielectric medium

• Express the force between two charges separated by distance r inside a medium with dielectric constant K.

Step 2: Write the force equation in air

• Express the force for the same charges separated by an equivalent distance r' in air.

Step 3: Equate both forces and solve 

• Set the two force equations equal to each other (since the interaction force is the same) and simplify to find that r' = r\[\sqrt{K}\].

Force After Adding Charges

Step 1: Find the initial force relationship

• Set up the initial Coulomb's law equation for charges \[+2\ \mu\text{C}\] and \[+4\ \mu\text{C}\] experiencing a \[20\text{ N}\] repulsive force, giving the constant term relation \[kr^2 = 2.5 \times 10^{12}\].

Step 2: Determine the new charges

• Add \[-6\ \mu\text{C}\] to both charges, resulting in new charges of \[-4\ \mu\text{C}\] and \[-2\ \mu\text{C}\].

Step 3: Calculate the new electrostatic force

• Substitute the new charges and the \[kr^2\] term back into the force formula to find the final force of \[20\text{ N}\].