Electrostatics – Stage 3 (Page 7)
Method of Images, Image Charges & Stability
1️⃣ Why Method of Images Exists (Philosophy First)
Instead of solving complex boundary-value problems, we replace conductors with imaginary (image) charges.
Key idea:
Electric field outside conductor must satisfy boundary conditions
If an imaginary system produces the SAME field, it is physically valid outside the conductor.
2️⃣ Most Fundamental Image System
Point charge near infinite grounded conducting plane
- Real charge: +q at distance d
- Image charge: −q at distance d behind the plane
✔ Plane itself disappears from calculations ✔ Boundary condition (V = 0) is automatically satisfied
3️⃣ Force on Charge Near Conducting Plane
Force on real charge is same as force due to image:
F = (1 / 4πϵ₀) · q² / (2d)²
⚠️ Direction: Always attraction
4️⃣ Energy of Charge Near Conductor (IIT Favorite)
Work done in bringing charge from infinity:
U = − (1 / 16πϵ₀) · q² / d
✔ Energy is NEGATIVE → bound system
5️⃣ Stability of Charge Near Conducting Plane
Equilibrium exists but is UNSTABLE
Reason:
- Force always attracts toward plane
- No restoring force parallel to surface
Earnshaw’s theorem is NOT violated — conductor provides constraint.
6️⃣ Charge Near Conducting Sphere (Grounded)
Image system:
- Image charge magnitude changes
- Image charge position shifts
Key relations:
q' = − q · (R / r)
r' = R² / r
✔ Image charge lies inside sphere
7️⃣ When NOT to Use Image Method
- Multiple irregular conductors
- Non-ideal boundaries
- Dielectric interfaces (usually)
⚠️ Image method is powerful but selective
8️⃣ Small Oscillations Using Image Charges
IIT trick:
- Replace conductor with image charge
- Find force for small displacement x
- Linearize force
Leads to:
F ≈ − k x → SHM
✔ Combines Electrostatics + SHM
9️⃣ Common IIT Traps
- Forgetting factor ½ in energy
- Wrong sign of image charge
- Assuming stability without checking second derivative
🔟 Master Insight (Gold)
Image charges are not real — but their effects on real charges ARE real.
➡️ Next Page
Stage 3 – Page 8:
Electrostatic Shielding, Cavities, Induced Charges & Field Inside Cavities
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