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

Given : $(\mathrm{b}-\mathrm{a})=1\times {10}^{-3}\mathrm{m}$         ….(i)

and $\mathrm{C}=4{\mathrm{\pi \epsilon }}_0\left(\frac{\mathrm{ab}}{\mathrm{b}-\mathrm{a}}\right)=1\times {10}^{-6}$

$\Rightarrow 1\times {10}^{-6}=\frac{1}{9\times {10}^9}\left(\frac{\mathrm{ab}}{{10}^{-3}}\right)$

$\Rightarrow \mathrm{ab}=9$                                  ….(ii)

From equations (i) and (ii) 

$\mathrm{b}-\frac{9}{\mathrm{b}}=\frac{1}{1000}\Rightarrow 1000{\mathrm{b}}^2-\mathrm{b}-9000=0$

$\Rightarrow \mathrm{b}=\frac{1\pm \sqrt{(-1{)}^2-4(1000\left)\right(-9000)}}{2\times 1000}$  {Solving of quadratic equation}

$\Rightarrow \mathrm{b}=\frac{1\pm \sqrt{36\times {10}^6}}{2000}\approx \frac{\sqrt{36\times {10}^6}}{2000}=3\mathrm{m}.$