Question

The capacitance of a spherical condenser is $1\, \mu F$. If the spacing between the two spheres is $1\, mm$, 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,
$r_{2}-r_{1}=1\, mm =10^{-3} m$
$\therefore C=4 \pi \varepsilon_{0} \frac{r_{1} r_{2}}{r_{2}-r_{1}}$
$\Rightarrow 10^{-6}=\frac{1}{9 \times 10^{9}} \times \frac{\left(r_{2}-10^{-3}\right) r_{2}}{10^{-3}}$
$\Rightarrow 9=r_{2}{ }^{2}-10^{-3} r_{2}$
$\Rightarrow 9000=1000 r_{2}{ }^{2}-r_{2}$
$\Rightarrow 1000 r_{2}{ }^{2}-r_{2}-9000=0$
$\Rightarrow r_{2}=\frac{-(-1) \pm \sqrt{(-1)^{2}-4 \times 1000 \times(-9000)}}{2 \times 1000}$
$=\frac{1 \pm \sqrt{36 \times 10^{6}}}{2000}=\frac{1 \pm 6000}{2000}$
$=\frac{1+6000}{2000}=3.0005 \simeq 3\, m$