Electrostatics – Stage 3 (Page 6)
Electrostatic Equilibrium, Stability & Oscillations
1️⃣ What Does Electrostatic Equilibrium Really Mean?
Electrostatic equilibrium means NO net force on charges.
For a charge to be in equilibrium:
Net electric field at its position = 0
Important:
- Equilibrium does NOT imply zero potential
- Only field must be zero
2️⃣ Types of Equilibrium (IIT Favorite)
| Type | Energy Condition | Behavior |
|---|---|---|
| Stable | U = minimum | Returns when disturbed |
| Unstable | U = maximum | Moves away |
| Neutral | U = constant | Stays anywhere |
3️⃣ Stability Condition Using Energy
At equilibrium point:
dU/dx = 0
For stability:
d²U/dx² > 0
✔ This test avoids force analysis completely
4️⃣ Earnshaw’s Theorem (Extremely Important)
No stable equilibrium of a charge is possible using only electrostatic forces
Consequences:
- You cannot trap a charge using static charges alone
- True stability requires non-electrostatic forces
⚠️ Magnetic fields, gravity, or motion can bypass this
5️⃣ Conductors in Electrostatic Equilibrium
- Electric field inside conductor = 0
- Potential is constant throughout
- Charge resides only on surface
Why?
Any internal field would cause charge motion → not equilibrium
6️⃣ Small Oscillations About Equilibrium
If a charge is slightly displaced from a stable point:
Restoring force ≈ linear
This leads to:
Simple Harmonic Motion (SHM)
General form:
F = − k x
7️⃣ SHM Frequency from Energy
Near equilibrium:
U ≈ U₀ + ½ k x²
Angular frequency:
ω = √(k / m)
✔ No force calculation needed
8️⃣ Electrostatic Oscillator (IIT Style)
Common IIT setup:
- Charge between two fixed charges
- Charge near conducting plane
Procedure:
- Find equilibrium position
- Expand force or energy for small x
- Identify SHM constant
9️⃣ Trap: Zero Field ≠ Stable Equilibrium
E = 0 does NOT guarantee stability
Check:
- Second derivative of energy
- Nature of force after displacement
🔟 IIT Examiner Checklist
✔ Use energy derivatives ✔ Remember Earnshaw’s theorem ✔ SHM near stable equilibrium ✔ Avoid force balance unless forced ✔ Stability = minimum energy
➡️ Next Page
Stage 3 – Page 7:
Advanced Applications – Image Charges, Method of Images & Stability Traps
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