Coulomb's Law — Revision Notes

⚡ 30-Second Revision

Coulomb's Law (Magnitude): — $F = k \frac{|q_1 q_2|}{r^2}$
Coulomb's Constant: — $k = \frac{1}{4piepsilon_0} \approx 9 \times 10^9\,\text{N m}^2/\text{C}^2$
Permittivity of Free Space: — $\epsilon_0 \approx 8.854 \times 10^{-12},\text{C}^2/\text{N m}^2$
Elementary Charge: — $e \approx 1.6 \times 10^{-19},\text{C}$
Force in Medium: — $F_{\text{medium}} = \frac{F_{\text{vacuum}}}{\epsilon_r}$, where $\epsilon_r$ is the dielectric constant.
Like charges repel, unlike charges attract.
Principle of Superposition: — Net force is vector sum of individual forces.

2-Minute Revision

Coulomb's Law is the fundamental law quantifying the electrostatic force between two stationary point charges. The force's magnitude is directly proportional to the product of the charges and inversely proportional to the square of the distance between them ($F = k \frac{|q_1 q_2|}{r^2}$).

The constant $k$ depends on the medium, being $1/(4piepsilon_0)$ in vacuum. The force is attractive for opposite charges and repulsive for like charges, always acting along the line joining them. When charges are in a material medium, the force is reduced by a factor equal to the medium's dielectric constant ($\epsilon_r$), so $F_{\text{medium}} = F_{\text{vacuum}}/\epsilon_r$.

For systems with multiple charges, the net force on any charge is found by vectorially adding the individual forces exerted by all other charges (Principle of Superposition). Remember to convert units to SI (Coulombs, meters) and handle vector addition carefully, often by resolving forces into components.

5-Minute Revision

Coulomb's Law is the bedrock of electrostatics, describing the interaction between stationary point charges. The magnitude of the force ($F$) is given by $F = k \frac{|q_1 q_2|}{r^2}$, where $q_1, q_2$ are charge magnitudes, $r$ is their separation, and $k$ is Coulomb's constant ($9 \times 10^9\,\text{N m}^2/\text{C}^2$ in vacuum). The direction is along the line joining the charges: repulsive for like charges, attractive for unlike charges.

Key Points for NEET:

1
Units: — Always convert charges to Coulombs ($1,\mu\text{C} = 10^{-6},\text{C}$) and distances to meters ($1\,\text{cm} = 10^{-2},\text{m}$).
2
Vector Nature: — For multiple charges, use the Principle of Superposition. Calculate each individual force (magnitude and direction) and then perform vector addition. For example, if two forces $\vec{F}_1$ and $\vec{F}_2$ act at an angle $\theta$, the resultant magnitude is $F_{\text{net}} = \sqrt{F_1^2 + F_2^2 + 2F_1 F_2 \cos\theta}$. For forces along axes, sum x-components and y-components separately.
3
Effect of Medium: — The force in a medium is $F_{\text{medium}} = F_{\text{vacuum}}/\epsilon_r$, where $\epsilon_r$ is the dielectric constant. $\epsilon_r \ge 1$, so force always decreases or stays the same.
4
Equilibrium Problems: — Set the net force on the charge(s) to zero. This often involves finding a position where opposing forces cancel out. For a charge $q$ to be in equilibrium between two identical charges $Q$, $q$ must be negative and placed at the midpoint.
5
Comparison with Gravity: — Both are inverse square laws. Electrostatic force is much stronger, can be attractive or repulsive, and depends on the medium. Gravitational force is always attractive and independent of the medium.

Example: Two charges $+2,\mu\text{C}$ and $-3,\mu\text{C}$ are $30\,\text{cm}$ apart. $q_1 = 2 \times 10^{-6},\text{C}$, $q_2 = -3 \times 10^{-6},\text{C}$, $r = 0.3\,\text{m}$. $F = (9 \times 10^9) \frac{|(2 \times 10^{-6})(-3 \times 10^{-6})|}{(0.3)^2} = (9 \times 10^9) \frac{6 \times 10^{-12}}{0.09} = (9 \times 10^9) \times (6 \times 10^{-12}) \times \frac{1}{0.09} = 54 \times 10^{-3} \times \frac{100}{9} = 6 \times 10^{-3} \times 100 = 0.6\,\text{N}$. The force is attractive.

Prelims Revision Notes

Coulomb's Law: — $F = k \frac{|q_1 q_2|}{r^2}$. This is the magnitude. Direction: like charges repel, unlike charges attract.
Coulomb's Constant ($k$): — In vacuum, $k = \frac{1}{4piepsilon_0} \approx 9 \times 10^9\,\text{N m}^2/\text{C}^2$. Use this value unless specified otherwise.
Permittivity of Free Space ($\epsilon_0$): — $8.854 \times 10^{-12},\text{C}^2/\text{N m}^2$.
Elementary Charge ($e$): — $1.602 \times 10^{-19},\text{C}$. All charges are integer multiples of $e$.
Units: — Always convert charges to Coulombs (C) and distances to meters (m) before calculations. $1,\mu\text{C} = 10^{-6},\text{C}$, $1\,\text{nC} = 10^{-9},\text{C}$, $1\,\text{cm} = 10^{-2},\text{m}$.
Vector Form: — $\vec{F}_{21} = k \frac{q_1 q_2}{r^3} \vec{r}_{12}$. Remember $\vec{r}_{12} = \vec{r}_2 - \vec{r}_1$.
Principle of Superposition: — Net force on a charge is the vector sum of forces from all other charges. $\vec{F}_{\text{net}} = \sum \vec{F}_i$. This means you must consider directions carefully. Use component method or parallelogram law for vector addition.
Effect of Medium: — When charges are in a medium with dielectric constant $\epsilon_r$, the force is $F_{\text{medium}} = \frac{F_{\text{vacuum}}}{\epsilon_r}$. Since $\epsilon_r \ge 1$, the force in a medium is always less than or equal to the force in vacuum.
Equilibrium: — For a charge to be in equilibrium, the net force on it must be zero. This often involves balancing attractive and repulsive forces. For a third charge $q$ to be in stable equilibrium between two identical charges $Q$, $q$ must be negative and placed exactly at the midpoint.
Limitations: — Coulomb's Law is valid for stationary point charges. It does not apply to charges in motion (where magnetic effects arise) or for distances less than $10^{-15},\text{m}$ (nuclear forces dominate).
Comparison with Gravity: — Both are inverse square laws. Electrostatic force is much stronger ($10^{39}$ times for electron-proton), can be attractive or repulsive, and depends on the medium. Gravitational force is always attractive and independent of the medium.

Vyyuha Quick Recall

Charges Often Undergo Large Opposing Movements Because Strength Largely Alters With Radius Squared. (Coulomb's Law: Like charges repel, unlike attract, force is proportional to product of charges and inversely proportional to square of radius.)