Question

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

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

Answer 1

Answer: (D)

$3m$

Answer 2

Given : (b – a) = 1 × 10^–3 m ..... (i)

and $C = 4\pi\varepsilon_{0}\left( \frac{ab}{b - a} \right) = 1 \times 10^{- 6}$

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

⇒ $ab = 9$ ..... (ii)

From equations (i) and (ii)

$b - \frac{9}{b} = \frac{1}{1000}$ ⇒ 1000 b² – b – 9000 = 0

⇒ $b = \frac{1 \pm \sqrt{( - 1)^{2} - 4(1000)( - 9000)}}{2 \times 1000}$

{Solving of quadratic equation}

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