A spherical capacitor consists of two concentric spherical conductors of inner one of radius $R_{1}$ maintained at potential $V_{1}$ and the outer one of radius $R_{2}$ at potential $V_{2}$ . The potential at a point p at a distance x from the centre $(R_{2} > x > R_{1})$ is:

\frac{V_{1} - V_{2}}{R_{2} - R_{1}} (x - R_{1})

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{V_{1} R_{1} (R_{2} - x) + V_{2} R_{2} (x - R_{1})}{x (R_{2} - R_{1})}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

V_{1} + \frac{V_{2} x}{R_{2} - R_{1}}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{V_{1} + V_{2}}{R_{2} + R_{1}} x

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $\frac{V_{1} R_{1} (R_{2} - x) + V_{2} R_{2} (x - R_{1})}{x (R_{2} - R_{1})}$
Let $q_{1}$ and $q_{2}$ be the charges on inner and outer sphere respectively.
Here, $V_{1} = k [\frac{q_{1}}{R_{1}} + \frac{q_{2}}{R_{2}}]$
or $V_{1} = k [\frac{q_{1} R_{2} + q_{2} R_{1}}{R_{1} R_{2}}]$ or $q_{2} = \frac{V_{1} R_{1} R_{2}}{k R_{1}} - \frac{q_{1} R_{2}}{R_{1}} = \frac{V_{1} R_{2}}{k} - \frac{q_{1} R_{2}}{R_{1}}$
and $V_{2} = k [\frac{q_{1} + q_{2}}{R_{2}}]$ or $V_{2} R_{2} = k [q_{1} + q_{2}]$
or $\frac{V_{2} R_{2}}{k} = [q_{1} + \frac{V_{1} R_{2}}{k} - \frac{q_{1} R_{2}}{R_{1}}]$
or $\frac{V_{2} R_{2}}{k} - \frac{V_{1} R_{2}}{k} = q_{1} [\frac{R_{1} - R_{2}}{R_{1}}]$
or $q_{1} = \frac{(V_{2} - V_{1}) R_{1} R_{2}}{k (R_{1} - R_{2})}$
now , $q_{2} = \frac{V_{1} R_{2}}{k} - \frac{(V_{2} - V_{1}) R_{1} R_{2}}{k (R_{1} - R_{2})} \times \frac{R_{2}}{R_{1}}$
now, $V_{x} = k [\frac{q_{1}}{x} + \frac{q_{2}}{R_{2}}]$
putting the value of $q_{1}$ and $q_{2}$ , we get
$V_{x} = \frac{V_{1} R_{1} (R_{2} - x) + V_{2} R_{2} (x - R_{1})}{(R_{2} - R_{1}) x}$

Suggest Corrections

Similar questions

Q. Two particles 'A' and 'B', move with constant velocities

_{1}

and

_{2}

respectively. At the initial moment, their position vectors are

_{1}

and

_{2}

respectively. The condition for particle 'A' and 'B' for their collision is

Q. The emf (V) and internal resistance (

r

) of a single battery which is equivalent to a parallel combination of two batteries of emf's

V_{1}

and

V_{2}

and internal resistance

r_{1}

and

r_{2}

, respectively, with polarities as shown in the figure are given by

\frac{V_{1} r_{2} - V_{2} r_{1}}{r_{1} + r_{2}}, \frac{r_{1} r_{2}}{r_{1} + r_{2}}

Two satellites of masses $m_{1}$ and $m_{2} (m_{1} > m_{2})$ are revolving round the earth in circular orbits of radius $r_{1}$ and $r_{2} (r_{1} > r_{2})$ respectively. Which of the following statements is true regarding their speeds $v_{1}$ and $v_{2}$ ?