🌟 Potential Energy of a System of Charges 🌟

1. Two–Charge System

Imagine two point charges q1 and q2 separated by a distance r12. The work you do to build this pair shows up as their electrostatic potential energy:

\( U \;=\; \dfrac{1}{4\pi\varepsilon_0}\,\dfrac{q_1\,q_2}{r_{12}} \) (Equation 2.22) 🔋:contentReference[oaicite:0]{index=0}

  • Like charges (q1q2 > 0) ➜ positive \(U\). You push them together, so you supply energy.⚡:contentReference[oaicite:1]{index=1}
  • Unlike charges (q1q2 < 0) ➜ negative \(U\). They attract, so the system can release energy.🎯:contentReference[oaicite:2]{index=2}
  • The value depends only on the current positions, not on the path you chose. 🛣️:contentReference[oaicite:3]{index=3}

2. Three–Charge System (and the idea for many charges)

Build the trio q1, q2, and q3 step-by-step. Add up the pairwise contributions and you get

\( U \;=\; \dfrac{1}{4\pi\varepsilon_0}\!\left[\dfrac{q_1q_2}{r_{12}} + \dfrac{q_1q_3}{r_{13}} + \dfrac{q_2q_3}{r_{23}}\right] \) (Equation 2.26) 🔋:contentReference[oaicite:4]{index=4}

For four or more charges, keep adding every unique pair in the same way. The conservative nature of the electrostatic force guarantees that the final expression stays the same no matter how you assemble the charges.💡:contentReference[oaicite:5]{index=5}

3. Worked Example: Four Charges on a Square 🟥

Place the charges +q, −q, +q, −q at the corners of a square ABCD (side =d). Moving the charges in one convenient order and totalling the work gives the system’s energy:

\( U \;=\; -\dfrac{q^{2}}{4\pi\varepsilon_0\,d}\,\bigl(4-2\sqrt{2}\bigr) \) 🔢:contentReference[oaicite:6]{index=6}

Bring an extra charge q0 to the square’s centre. The potentials from the four corners cancel, so you do zero extra work! 🎉:contentReference[oaicite:7]{index=7}

4. Quick Facts to Remember 🚀

  • Potential energy depends on products of charges and their separations, not on the order of assembly.:contentReference[oaicite:8]{index=8}
  • Positive \(U\) signals stored energy due to repulsion; negative \(U\) signals energy release due to attraction.:contentReference[oaicite:9]{index=9}
  • Add energies for every unique pair to handle multiple charges.:contentReference[oaicite:10]{index=10}
  • The electrostatic force is conservative, so path choice never changes the answer.:contentReference[oaicite:11]{index=11}

🧠 High-Yield Ideas for NEET

  1. Pair Formula Mastery: Know \( U = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}} \) and apply sign logic quickly.
  2. Add-All-Pairs Rule: For three or more charges, sum every pair once – a frequent NEET trick.
  3. Path Independence: Work done equals final energy only; route never matters (often tested conceptually).
  4. Zero-Potential Regions: Symmetry can make potential (and added work) vanish, as in the square-centre example.

✨ Keep practicing these concepts—you’ve got this! ✨