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If we have a spherical capacitor with inner radius or r1 and outer radius of r2, with charges (+/-)q on them and there is a dielectric material (with constant e) in between them with.

What kind of a potential would this create outside the entire capacitor? in the region with the dielectric? and inside the entire thing?

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3 Answers 3

Using the Gauss's law (see this Link), the solution is as follow, $$ \Phi (r) = \left\{ {\begin{array}{*{20}{c}} {\frac{q}{{4\pi \varepsilon {r_1}}}\,\,\,\,for\,\,\,r \le {r_1}}\\ {\frac{q}{{4\pi \varepsilon r}}\,\,\,\,for\,\,\,{r_1} \le r \le {r_2}}\\ {0\,\,\,\,\,\,\,for\,\,\,r \ge {r_2}} \end{array}} \right. $$

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You can write down these fields directly without calculating anything because the field of a charged sphere is the same as that of a point charge outside the sphere and zero inside the sphere.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/potsph.html

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This is quite simple as this is a standard equation for the capacitance. You will have

$$C=\epsilon C_0$$

being $C_0$ the capacitance in vacuum. For a full set of formulas in different geometrical configurations see here.

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