Master Electric Fields and Capacitors in JEE Electrodynamics

Apex Class • IITJEE Electrodynamics Lab

Electric Field (E), Potential (V) Graphs & Capacitor Dielectrics Lab

Compare Hollow vs. Solid Sphere E & V Graphs and master Dielectric Insertion (Battery Connected vs. Disconnected)!

🌐 Select a Charge Distribution to Inspect E(r) and V(r) simultaneously:

1. Hollow Conducting Sphere 2. Solid Non-Conducting Sphere 3. Charged Ring on Axis (x = R/√2) 4. Electric Dipole (Axial vs Equatorial)

Internal Field (r < R)

E_in = 0 (Zero!) E_in = kQr / R³ (∝ r) At Center (x=0): E = 0 E_axial = 2kp / r³

Gauss’s Law Application

Internal Potential (r < R)

V_in = kQ / R (Const!) V_center = 1.5 (kQ/R) V_x = kQ / √(R² + x²) V_axial = kp/r² | V_eq = 0

E = -dV / dr Relation

Maximum Field Value

kQ / R² (At Surface) kQ / R² (At Surface) 2kQ / (3√3 R²) |E_axial| = 2 |E_eq|

Peak Location Check

Self-Energy / Torque

U = kQ² / (2R) U = (3/5) kQ² / R Small osc: ω = √(kQq/mR³) τ = p × E | U = -p · E

High-Yield JEE Formula

🔋 Insert Dielectric Slab (K = 3) into Parallel Plate Capacitor. Select Circuit State:

1. Air Capacitor (C₀ = ε₀A/d) 2. Battery DISCONNECTED (Q = Const!) 3. Battery CONNECTED (V = Const!) 4. Partial Slab of Thickness t (t < d)

Capacitance (C)

C₀ = ε₀ A / d C = K C₀ (Increases!) C = K C₀ (Increases!) ε₀A / (d – t + t/K)

Always Increases with K

Plate Charge (Q)

Q₀ = C₀ V₀ Q = Q₀ (Constant!) Q = K Q₀ (Increases K×) Q_ind = Q (1 – 1/K)

Conservation of Charge

Voltage (V) & Field (E)

V₀ | E₀ = V₀ / d V₀ / K | E₀ / K (Drops!) V₀ | E₀ (Constant!) E_slab = E₀ / K

V = E × d Relation

Stored Energy (U)

U₀ = ½ C₀ V₀² U = U₀ / K (Decreases!) U = K U₀ (Increases!) U = Q² / (2C)

Use Q²/2C when Q=Const; ½CV² when V=Const