A metallic sphere of radius R is surrounded by a concentric dielectric shell of inner radius R, and outer radius 3R/2. This is surrounded by a concentric, thin, metallic shell of radius 2R (see Figure 10). The dielectric constant of the shell is [kappa]. What is the capacitance of this contraption ?
Suppose the charge on the inner sphere is Qfree. The electric field inside the dielectric can be determined by applying Gauss' law for a dielectric (eq.(50)) and using as the integration volume a sphere of radius r (where R < r < 3R/2)
The electric field in this region is therefore given by
The electric field in the region between 3R/2 and 2R can be obtained in a similar manner, and is equal to
Using the electric field from eq.(52) and eq.(53) we can determine the potential difference [Delta]V between the inner and outer sphere:
The capacitance of the system can be obtained from eq.(54) using the definition of the capacitance in terms of the charge Q and the potential difference [Delta]V:
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