← Home / Electric Potential & Flux

Potential & Flux Density

Lectures 5–6 · Tutorial 4  |  Voltage, Work, D-field, Capacitors, Dielectrics

🔋 Electric Potential V

Work Done Moving a Charge — Path from A to B

+Q E → A r_A (far) B r_B (closer) path (any path works!) V_AB = V_B − V_A = −∫[A→B] E·dl (J/C = Volts)

Electric Potential Definition

Work done per unit charge to move a test charge from A to B against the E field.

V_AB = W/q = −∫[A→B] E·dl [V]
V_AB = V_B − V_A

Potential from a Point Charge

Reference at infinity (V=0 at r=∞):

V = Q / (4πε₀r)

This is the work done to bring +1C from ∞ to distance r from Q.

Relation Between E and V

Electric field is the negative gradient of potential.

E = −∇V
Ex = −∂V/∂x
Ey = −∂V/∂y
Ez = −∂V/∂z
💡 Between parallel plates: |E| = |ΔV|/d

Conservative Field Property

For electrostatic fields: Work around a closed loop = 0.

∮ E·dl = 0 (closed loop)

This means V_BA = −V_AB. The path taken doesn't affect ΔV.

📝 Tutorial Example
q₁=2μC at origin, q₂=−6μC at (0,3). Find V at P(4,0) and energy to bring q₃=3μC from ∞ to P.
r₁ = 4m, r₂ = √(16+9) = 5m
V = q₁/(4πε₀×4) + q₂/(4πε₀×5) = −6.29×10³ V
W = q₃×V = 3×10⁻⁶ × (−6290) = −18.87×10⁻³ J
📡 Electric Flux Density D

Flux Density D

D is independent of the material — depends only on the source charge, not ε.

D = ε·E = Q/(4πr²) · â_r [C/m²]

D vs E

D = ε₀·εᵣ·E = ε·E

In free space: D = ε₀·E. In a dielectric: D = ε₀εᵣE. D formulas use the same Coulomb form as E — just replace ε₀ with ε.

Gauss's Law (Differential)

∇·D = ρᵥ [C/m³]

Divergence of D equals the volume charge density. This is one of Maxwell's equations.

📝 Example — D from point charge
12 nC at origin. Find D at P(0, −3, 2).
r = (0, −3, 2) → |r| = √13
D = 12×10⁻⁹ × (−3â_y + 2â_z) / (4π × 13^(3/2))
D = −61.12â_y + 70.75â_z pC/m²
🔋 Capacitors

Parallel-Plate Capacitor Schematic

+ + + + + + + + − − − − − − − − εᵣ (dielectric) E → d Area = A C = ε₀εᵣA/d | Q = CV | E_stored = ½CV²

Capacitance Fundamentals

C = Q/V = ε₀εᵣA/d [F]
E_stored = ½CV² [J]

C increases with: larger area A, higher εᵣ, smaller gap d.

Series Capacitors

Same charge Q on each; voltages add up.

1/C_eq = 1/C₁ + 1/C₂ + 1/C₃

C_eq is always less than smallest individual C.

Parallel Capacitors

Same voltage V across each; charges add up.

C_eq = C₁ + C₂ + C₃

C_eq is always greater than largest individual C.

Dielectric Advantages

  • Increases capacitance (C ∝ εᵣ)
  • Increases max operating voltage
  • Provides mechanical support between plates
  • Allows smaller d (higher C)

Dielectric Breakdown

Maximum E field the dielectric can withstand before conducting. Breakdown voltage = maximum safe voltage.

💡 Keyboard example: Pressing a key compresses d → C = ε₀εᵣA/d increases. Answer: capacitance INCREASES.

Charging vs Discharging

Charging: Capacitor acts as load, stores energy, current decreases to zero.
Discharging: Capacitor acts as source, releases energy, voltage drops.