Question

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

• A. 30 cm
• B. 6 cm
• C. 5 cm
• D. 3 cm

Answer 1

Answer: (D)

Given, capacitance of spherical capacitor,
$C=1\mu F=10^{-6}F$
Spacing between two spheres of spherical capacitor,
$r_2-r_1=1mm=10^{-3}m$
$\therefore C=4\pi {\epsilon }_0\frac{r_1{r}_2}{r_2-r_1}$
$\Rightarrow 10^{-6}=\frac{1}{9\times 10^9}\times \frac{\left(r_2-10^{-3}\right)r_2}{10^{-3}}$
$\Rightarrow 9=r_2{}^2-10^{-3}{r}_2$
$\Rightarrow 9000=1000r_2{}^2-r_2$
$\Rightarrow 1000r_2{}^2-r_2-9000=0$
$\Rightarrow r_2=\frac{-(-1)\pm \sqrt{(-1{)}^2-4\times 1000\times (-9000)}}{2\times 1000}$
$=\frac{1\pm \sqrt{36\times 10^6}}{2000}=\frac{1\pm 6000}{2000}$
$=\frac{1+6000}{2000}=3.0005\simeq 3m$