Space between plates of a parallel plate capacitor is filled by a weakly conducting viscous fluid of resistivity $ρ$ and dielectric constant K. Distance between the plates is d. The capacitor is connected across an ideal battery of terminal voltage V and a small conducting and almost massless ball having a charge $q_{0}$ is released in the fluid, force of viscous drag on the ball in the fluid is $F = β υ$ , where $υ$ is velocity of the ball and $β$ is a known factor. If the ball stops before striking any of the plates, the maximum distance ball will move is ____ (assume no gravity)

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$\frac{ρ ε_{0} q_{0} V}{β d}$

$\frac{ρ K ε_{0} q_{0} V}{β d}$

$\frac{ρ ε_{0} q_{0} V}{K β d}$

$\frac{ρ ε_{0} q_{0} V}{2 K β d}$

answer is B.

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Detailed Solution

As the ball moves, charge on the ball leaks, since the medium is conducting the charge on the ball gets drifted due to potential difference
$q E - β υ = m \frac{d υ}{d t} E \int_{0}^{\infty} q d t - β \int_{0}^{\infty} υ d t = \int_{0}^{0} m d υ E \int_{0}^{\infty} q_{0} e^{- \frac{t}{τ}} d t - β x = 0 q_{0} E τ = β x x = \frac{q_{0} E τ}{β} = \frac{q_{0} V ρ K ε_{0}}{β d} [h e r e τ = ρ K ε_{0}]$