A spherical capacitor is made of two spherica | vedclass

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(C) Case $1$: When the outer shell of radius $b$ is grounded,the capacitance $C_1$ is given by the formula for a spherical capacitor: $C_1 = 4\pi \var

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$4\pi \varepsilon_0 b$

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(C) Case $1$: When the outer shell of radius $b$ is grounded,the capacitance $C_1$ is given by the formula for a spherical capacitor: $C_1 = 4\pi \varepsilon_0 \frac{ab}{b-a}$.Case $2$: When the inner shell of radius $a$ is grounded,the system acts as two capacitors in parallel: the inner shell (isolated) and the outer shell (acting as a spherical conductor of radius $b$). The total capacitance $C_2$ is the sum of the capacitance of the spherical shell $b$ $(4\pi \varepsilon_0 b)$ and the mutual capacitance between the shells $(4\pi \varepsilon_0 \frac{ab}{b-a})$. Thus,$C_2 = 4\pi \varepsilon_0 b + 4\pi \varepsilon_0 \frac{ab}{b-a} = 4\pi \varepsilon_0 \left( \frac{b^2 - ab + ab}{b-a} \right) = 4\pi \varepsilon_0 \frac{b^2}{b-a}$.The difference is $C_2 - C_1 = 4\pi \varepsilon_0 \left( \frac{b^2}{b-a} - \frac{ab}{b-a} \right) = 4\pi \varepsilon_0 \left( \frac{b(b-a)}{b-a} \right) = 4\pi \varepsilon_0 b$.

$A$ spherical capacitor is made of two spherical shells of radii $a$ and $b$ $(b > a)$. The medium between the shells is air. What is the difference between the capacitances when the outer shell is grounded and when the inner shell is grounded?

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