EQS Solution with Uniformly Polarized Sphere

Editor：王宥翔

Adviser：江簡富 教授

    Denote the potentials inside and outside of the sphere as Φ_i  and Φ_o, respectively. The boundary conditions require that  Φ_o − Φ_i = 0 and − ε₀(∂Φ_o /∂r) + ε₀(∂Φ_i /∂r) = P₀ cos θ. Physical consideration also demands that Φ_o (r → ∞) = 0, and  Φ_i (r = 0) is finite. The functional form of surface polarization charge density and the physical consideration suggest the forms of potential distributions as

and

Furthermore, we can find the E_o and E_i from Φ_o and Φ_i. Thus,

and

(1) Potential Distribution

(2) Electric Field Distribution