The capacitance of a spherical condenser is $1 \mu \mathrm{F}$. If the spacing between the two spheres is $1 \mathrm{~mm}$, the radius of the outer sphere is

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Aicharya · Accepted Answer

Given, capacitance of spherical capacitor, $C=1 \mu \mathrm{F}=10^{-6} \mathrm{~F}$ Spacing between two spheres of spherical capacitor, $r_2-r_1=1 \mathrm{~mm}=10^{-3} \mathrm{~m}$ $\begin{array}{ll} \therefore & C=4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2-r_1} \ \Rightarrow & 10^{-6}=\frac{1}{9 \times 10^9} \times \frac{\left(r_2-10^{-3}\right) r_2}{10^{-3}} \ \Rightarrow & 9=r_2^2-10^{-3} r_2 \Rightarrow 9000=1000 r_2^2-r_2 \ \Rightarrow & 1000 r_2^2-r_2-9000=0 \end{array}$ $\begin{aligned} \Rightarrow r_2 & =\frac{-(-1) \pm \sqrt{(-1)^2-4 \times 1000 \times(-9000)}}{2 \times 1000} \ & =\frac{1 \pm \sqrt{36 \times 10^6}}{2000}=\frac{1 \pm 6000}{2000} \ & =\frac{1+6000}{2000}=3.0005 \approx 3 \mathrm{~m} \end{aligned}$

The capacitance of a spherical condenser is \(1 \mu \mathrm{F}\). If the spacing between the two spheres is…

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