Energy Distribution in Charged Spherical Systems - Edubirdie

University:
Santa Fe College
Course:
ETS 2527C | Electromechanical Components and Mechanisms
Academic year:

2024
Views:

183

Pages:

2
Author:

Jabin F.

Path Energy U = ε₀/2 ∫E²dτ U = ε₀/2 (∫₀ᴿ Ein² * 4πr²dr + ∫ᴿ∞ Eout² * 4πr²dr) U = ε₀/2 ∫₀ᴿ (q²/4πε₀r²)² * 4πr²dr + ε₀/2 ∫ᴿ∞ (q²/4πε₀r²)² * 4πr²dr = ε₀q²/8π²ε₀² ∫₀ᴿ r⁻² dr + ε₀q²/8π²ε₀² ∫ᴿ∞ r⁻² dr = ε₀q²/8π²ε₀² (-1/R + 1/R) = q²/8π²ε₀R U = q²/8π²ε₀R Usphere / Ushell = 6/5 Same radius Same charge f(r) = { K/r², r < R; 0, r > R } K is constant. Inside (r < R): ∫ρdτ = ∫₀ᴿ Kr² * 4πr²dr q = 4πK/5 R⁵ => K = 5q / 4πR⁵ ρ = 5φr⁴ / 4πR⁵ qenc = ∫₀ᴿ (5φr⁴ / 4πR⁵) * 4πr²dr = φr⁵ / R⁵ Ein = qenc / ε₀R * 4πr² = φr⁵ / ε₀R⁵ * 4πr² = 4φr³ / 4πε₀R⁵ → for a spherical symmetric charge distribution. # If ρ ∝ rn for a sphere Ein ∝ rn+1 Eout ∝ 1/r²

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