A spherical capacitor with varying permittivity of the dielectricIn general, capacitance calculations can be quite cumbersome involving complicated integrals. Whenever symmetries are present, we may find the capacitances much easier. Learn in this problem how to determine the properties of a spherical capacitor with a varying parmittivity of the dielectric.

Problem Statement

Schematic of a spherical capacitor with varying permittivity of the dielectric.Consider a spherical capacitor with inner and outer radii \(R_i\) and \(R_o\), respectively. Inside the metallic shells there is a dielectric that with a permittivity \(\varepsilon\) that may vary with respect to both angles \(\varphi\) and \(\theta\). \(\varepsilon\) may not change with the radius \(r\). What is the capacitance of such a device? You may want to proceed as follows:

  • Calculate first the capacitance when the permittivity is given by \( \varepsilon\left(\theta,\varphi\right) = \varepsilon_0\left(1+\varepsilon_\varphi \sin^2\varphi\right)\). Compare your result to a spherical capacitor with constant dielectric \(\varepsilon\),\[C=4\pi\varepsilon_0\varepsilon_{r}\frac{R_i R_o}{R_o-R_i}\ .\]
  • Now generalize your calculation for \(\varepsilon\left(\theta,\varphi \right)\). Explain what would be the conceptual difference if the permittivity could also vary on the radius.

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