Forces Between Multiple Charges ⚡

Coulomb’s law tells us how two point-charges push or pull each other. But in real life, a charge is usually surrounded by many other charges. How do we find the total force then? The answer is the principle of superposition—one of the most useful ideas in electrostatics. 🤓 :contentReference[oaicite:0]{index=0}

1 · Principle of Superposition 🤔

  • Statement: The total electric force on a charge \(q_1\) equals the vector sum of the individual Coulomb forces exerted by every other charge, taken one at a time. Other charges do not alter each individual force. :contentReference[oaicite:1]{index=1}
  • Force between two specific charges \( \displaystyle \mathbf F_{12}= \frac{1}{4\pi\varepsilon_0}\, \frac{q_1 q_2}{r_{12}^{\,2}}\;\hat{\mathbf r}_{12} \) :contentReference[oaicite:2]{index=2}
  • Total force on \(q_1\) caused by \(n-1\) neighbors \( \displaystyle \mathbf F_1=\sum_{i=2}^{n} \frac{1}{4\pi\varepsilon_0}\, \frac{q_1 q_i}{r_{1i}^{\,2}}\;\hat{\mathbf r}_{1i} \) (Eq. 1.5)
  • Why vectors? Each force points along the line joining the charges, so we must use head-to-tail (parallelogram) addition to mix directions correctly. 🧮 :contentReference[oaicite:3]{index=3}

2 · Quick Three-Charge Demo

For three charges \(q_1,q_2,q_3\), the net force on \(q_1\) is \( \mathbf F_{1}=\mathbf F_{12}+\mathbf F_{13} \). The same trick repeats for any charge you pick. :contentReference[oaicite:4]{index=4}

3 · Example 1.5 — Centroid Magic 🎯

Setup: Three equal charges \(q\) sit at the corners of an equilateral triangle of side \(l\). A fourth charge \(Q\) (same sign as \(q\)) rests at the triangle’s centroid.

  • Distances: Each vertex–centroid distance is \(l/\sqrt{3}\).
  • Forces from the three vertices have the same size \( \displaystyle F=\frac{3}{4\pi\varepsilon_0}\frac{Qq}{l^{2}} \).
  • Vector directions cancel perfectly by symmetry, so the net force on \(Q\) is zero. 🥳

4 · Example 1.6 — Two \(q\) and One –\(q\) 🔄

Setup: Charges \(q\), \(q\), and \(-q\) occupy the vertices of an equilateral triangle.

  • Each pair force magnitude \( \displaystyle F=\frac{q^{2}}{4\pi\varepsilon_0\,l^{2}} \).
  • Using the parallelogram rule, the net force on each charge has size \(F\) and points along a side (for the \(q\)’s) or along the angle-bisector (for \(-q\)).
  • The three forces themselves add up to \(\mathbf 0\), echoing Newton’s third law. ⚖️ :contentReference[oaicite:5]{index=5}

5 · Touch-and-Share Illustration 🪄

Want to see redistribution in action? In a classic experiment two identical metal spheres A and B carry charges \(q\) and \(q’\). Touching each with an uncharged twin splits their charges in half. Bringing A and B closer (half the distance) leaves the electric push unchanged because both charge strengths and the distance scale balance out! :contentReference[oaicite:6]{index=6}

6 · Key Takeaways 📌

  • Superposition lets you treat one pair at a time, then simply add.
  • Coulomb force stays untouched by other nearby charges.
  • Symmetry arguments often turn nasty sums into neat zeros.
  • Real-world problems rarely need all the algebra once you spot patterns. 😉

Important Concepts for NEET 🎓

  1. Principle of superposition—statement, vector nature, and general formula. :contentReference[oaicite:7]{index=7}
  2. Net force at a symmetric point (centroid of an equilateral triangle) ends up zero—great for quick MCQs. :contentReference[oaicite:8]{index=8}
  3. Force balance with mixed signs (two \(q\) and one \(-q\)) tests both superposition and direction skills. :contentReference[oaicite:9]{index=9}
  4. Charge redistribution on identical conductors and its effect on force. :contentReference[oaicite:10]{index=10}
  5. Connection to Newton’s third law—vector sum of internal forces can vanish in a closed charge system. :contentReference[oaicite:11]{index=11}