Classical Electrodynamics · Problem 3.27

Multipoles, Electrostatics of Macroscopic Media, Dielectrics — John David Jackson

Physics — E&M · neumann-green-function-spherical

Problem

Apply the Neumann Green function of Problem 3.26 to the situation in which the normal electric field is Er=−E0cos⁡θE_r = -E_0\cos\theta at the outer surface (r=br = b) and is Er=0E_r = 0 on the inner surface (r=ar = a). (a) Show that the electrostatic potential inside the volume VV is Φ(x)=E0rcos⁡θ1−p3(1+a32r3)\Phi(\mathbf{x}) = E_0 \frac{r\cos\theta}{1 - p^3}\left(1 + \frac{a^3}{2r^3}\right) where p=a/bp = a/b. Find the components of the electric field, Er(r,θ)=−E0cos⁡θ1−p3(1−a3r3),Eθ(r,θ)=E0sin⁡θ1−p3(1+a32r3)E_r(r,\theta) = -E_0\frac{\cos\theta}{1-p^3}\left(1 - \frac{a^3}{r^3}\right), \qquad E_\theta(r,\theta) = E_0\frac{\sin\theta}{1-p^3}\left(1 + \frac{a^3}{2r^3}\right) (b) Calculate the Cartesian or cylindrical components of the field, EzE_z and EρE_\rho, and make a sketch or computer plot of the lines of electric force for a typical case of p=0.5p = 0.5.

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