Question
Suppose the space between the two inner shells of the previous problem is filled with a dielectric of dielectric coastant K. Find the capacitance of the system between A and B.
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Answer
Here we should consider two spherical capacitor of capacitance
cab and cbc in series
$\text{Cab}=\frac{4\pi\in_0\text{abk}}{(\text{b}-\text{a})}$
$\text{Cbc}=\frac{4\pi\in_0\text{bc}}{(\text{c}-\text{b})}$
$\frac{1}{\text{C}}=\frac{1}{\text{Cab}}+\frac{1}{\text{Cbc}}$
$=\frac{(\text{b}-\text{a})}{4\pi\in_0\text{abk}}+\frac{(\text{c}-\text{b})}{4\pi\in_0\text{bc}}$
$=\frac{\text{c}(\text{b}-\text{a})+\text{ka}(\text{c}-\text{b})}{\text{k}4\pi\in_0\text{abc}}$
$\text{C}=\frac{\text{k}4\pi\in_0\text{abc}}{\text{c}(\text{b}-\text{a})+\text{ka}(\text{c}-\text{b})}$
cab and cbc in series
$\text{Cab}=\frac{4\pi\in_0\text{abk}}{(\text{b}-\text{a})}$
$\text{Cbc}=\frac{4\pi\in_0\text{bc}}{(\text{c}-\text{b})}$
$\frac{1}{\text{C}}=\frac{1}{\text{Cab}}+\frac{1}{\text{Cbc}}$
$=\frac{(\text{b}-\text{a})}{4\pi\in_0\text{abk}}+\frac{(\text{c}-\text{b})}{4\pi\in_0\text{bc}}$
$=\frac{\text{c}(\text{b}-\text{a})+\text{ka}(\text{c}-\text{b})}{\text{k}4\pi\in_0\text{abc}}$
$\text{C}=\frac{\text{k}4\pi\in_0\text{abc}}{\text{c}(\text{b}-\text{a})+\text{ka}(\text{c}-\text{b})}$
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