Question

A leaky parallel plate capacitor is filled completely with a material having dielectric constant K = 5 and electric conductivity σ = 7.4 × 10^-12 Ω^-1 m^-1. If the charge on the plate at the instant t = 0 is q = 8.85 µC, then the leakage current at the instant t = 12 s is ______ × 10^-1 µA.

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Answer 1

A leaky parallel plate capacitor is filled completely with a material having dielectric constant K = 5 and electric conductivity σ = 7.4 × 10^-12 Ω^-1 m^-1. If the charge on the plate at the instant t = 0 is q = 8.85 µC, then the leakage current at the instant t = 12 s is 2 × 10^-1 µA.

Explanation:

As in case of discharging of a capacitor through a resistance

q = q₀e^-t/CR

i = `-"dq"/"dt" = q_0/"CR"e^(-"t"//"CR")`

Here, CR = `((epsilon_0"KA")/"d")(rho"d"/"A") = (epsilon_0"K")/sigma` [as ρ = 1/σ]

i.e., CR = `(8.846xx10^-12xx5)/(7.4xx10^-12)` = 6

So, i = `(8.85xx10^-6)/6 e^(-12//6)`

= `(8.85xx10^-6)/(6xx7.39)` [As e = 2.718, e² = 7.39]

= 0.20

= 2 × 10^-1 µA