The capacitance of an isolated charged sphere is independent of both the charge on the sphere and the potential difference, it is dependent only on its radius
The capacitance of an isolated sphere is calculated as follows;
- let the charge on the sphere = Q
- let the potential difference = V
- let the radius of the sphere = R
The potential difference is given as;
[tex]V = \frac{kQ}{R}[/tex]
where;
k is Coulomb's constant [tex]= 4\pi \epsilon _0[/tex]
The capacitance is given as;
[tex]c = \frac{Q}{\Delta V} = \frac{Q}{V_1 - V_2} \\\\at \ infinite \ radius \ R , V_2 = 0\\\\c = \frac{Q}{V_1 } \ \ \ \ (V_1 = V = \frac{kQ}{R} )\\\\c = \frac{Q}{kQ/R} \\\\c = \frac{R}{k}\\\\c = \frac{R}{4\pi \epsilon_0}[/tex]
From the equation above, the capacitance (c) is directly proportional to its radius (R) and independent of both the charge (Q) on the sphere and the potential difference (V).
Thus, the capacitance of an isolated charged sphere is independent of both the charge on the sphere and the potential difference, it is dependent only on its radius
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