Analytic solutions for the harmonic potentials involving concentric layered dielectric spheres
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The mathematical problem of a conducting spherical core of radius 𝑎 concentrically covered by a dielectric phase of radius 𝑏 placed in an arbitrary external electric field is investigated. The vector field equations for the electric field (Maxwell equations) and the boundary conditions are transformed to a scalar boundary value problem (BVP) in terms of the harmonic potential functions. The harmonic potentials denoted by Φ𝐼(𝑟,𝜃,𝜙) and Φ𝐼𝐼(𝑟,𝜃,𝜙) where (𝑟,𝜃,𝜙) are spherical coordinates, satisfy the Laplace equations in the regions 𝑏<𝑟 and 𝑎 < 𝑟 < 𝑏, respectively. General analytical solutions for the potentials in the two phases are determined in infinite series form using spherical harmonics methods. Exact closed form solutions are also derived via an alternative approach. The latter solutions contain integrals involving harmonic functions. Our general solutions are applicable for arbitrary external potentials disturbed by a conducting spherical core with a dielectric coating. Several illustrative examples are investigated and exact solutions for them are constructed using our general solutions. The non-dimensional parameter 𝑘= 𝜀𝐼/(𝜀𝐼+𝜀𝐼𝐼) , where 𝜀𝐼 is the dielectric constant for the region 𝑟 > 𝑏 and 𝜀𝐼𝐼 is the dielectric constant for the region 𝑎 < 𝑟 < 𝑏, influences the potential patterns in the case of externally imposed constant and linear fields. Our results for the source induced field indicate that the force is positive or negative depending on 𝑘 < 0.5 or 𝑘 > 0.5. Furthermore, the force is greater than zero when the core radius 𝑎 approaches the value of the outer radius 𝑏. We believe that our mathematical results are of interest where coated dielectric objects are exposed to external electric fields.
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