⚡️ Coulomb’s Law — The Basics

When two point charges (tiny compared to the distance between them) interact, the electric force between them depends on just three things: the size of each charge, how far apart they are, and the straight-line direction connecting them. 🎯:contentReference[oaicite:0]{index=0}

📏 Mathematical Form

\( F = k \dfrac{q_{1}q_{2}}{r^{2}} \)   (attraction or repulsion along the line joining the charges)

  • \( q_{1}, q_{2} \) = the two charges (C)
  • \( r \) = distance between their centres (m)
  • \( k = 9 \times 10^{9}\,\text{N·m}^2\!/\,\text{C}^2 \) in SI units

Because \( k = \dfrac{1}{4\pi\varepsilon_{0}} \), many textbooks prefer the “epsilon-naught” version: \( \varepsilon_{0} = 8.854 \times 10^{-12}\,\text{C}^2\text{N}^{-1}\text{m}^{-2} \). ⚙️:contentReference[oaicite:1]{index=1}

🔭 Defining the Coulomb

Place two identical \( 1\;\text{C} \) charges exactly \( 1\;\text{m} \) apart in vacuum. They push each other with a whopping \( 9 \times 10^{9}\;\text{N} \) — that’s about a million tonnes!! 🚀 (No wonder we normally use milli- or micro-coulombs.) :contentReference[oaicite:2]{index=2}

🧭 Vector Form & Direction

\( \mathbf{F}_{21} = \dfrac{1}{4\pi\varepsilon_{0}}\,\dfrac{q_{1}q_{2}}{r_{21}^{\,2}}\;\hat{\mathbf{r}}_{21} \)

  • \( \mathbf{F}_{21} \): force on charge 2 by charge 1
  • \( \hat{\mathbf{r}}_{21} \): unit vector pointing from 1 to 2
  • If \( q_{1} \) and \( q_{2} \) have the same sign, the force is ⬅️ repulsive; if opposite, ➡️ attractive.
  • Notice \( \mathbf{F}_{12} = -\mathbf{F}_{21} \) — Newton’s third law in action!

The formula above works equally well for like and unlike charges; the sign of \( q_{1}q_{2} \) automatically flips the direction. 🔄:contentReference[oaicite:3]{index=3}

🌍 Electric vs Gravity — A Strength Check!

  • Electron-proton pair: \( \dfrac{F_{\text{electric}}}{F_{\text{gravity}}} \approx 2.4 \times 10^{39} \) 🤯
  • Two protons: \( \dfrac{F_{\text{electric}}}{F_{\text{gravity}}} \approx 1.3 \times 10^{36} \)

Electric forces absolutely dominate at atomic scales; gravity is practically invisible here. :contentReference[oaicite:4]{index=4}

🚀 Quick Acceleration Example

At a separation of \( 1\;\text{Å} = 10^{-10}\;\text{m} \)…

  • \( |F| = 2.3 \times 10^{-8}\;\text{N} \)
  • Electron: \( a_e \approx 2.5 \times 10^{22}\;\text{m/s}^2 \)
  • Proton : \( a_p \approx 1.4 \times 10^{19}\;\text{m/s}^2 \)

That’s why electrons zoom around so easily while gravity barely tugs on them! ⚡️:contentReference[oaicite:5]{index=5}

🎲 Sharing Charge by Contact (Thought Experiment)

Touch a charged sphere to an identical neutral sphere ➡️ charge splits equally. Repeat the process and you get \( q, \tfrac{q}{2}, \tfrac{q}{4}, … \). Coulomb cleverly used this trick with a torsion balance to test the inverse-square law. 🧪:contentReference[oaicite:6]{index=6}

🔗 Principle of Superposition

Need the force on one charge surrounded by many others? Easy! Just add the individual forces vectorially. Each pair-wise force is unchanged by the presence of the rest. \( \mathbf{F}_{\text{net}} = \sum_{i\neq1} \mathbf{F}_{1i} \). ➕➕➕:contentReference[oaicite:7]{index=7}

🧠 High-Yield NEET Nuggets

  1. Inverse-square dependence \( (F \propto 1/r^{2}) \) and product of charges \( (q_{1}q_{2}) \).
  2. Vector form with unit vector \( \hat{\mathbf{r}} \) — essential for multi-charge problems.
  3. Conversion \( k = 1/4\pi\varepsilon_{0} \) and the numerical value of \( \varepsilon_{0} \).
  4. Principle of superposition for systems of >2 charges.
  5. Electrical vs gravitational force comparison (orders of magnitude!).

🌟 Happy studying & keep the sparks flying! 🌟