The material filled between the plates of a parallel plate capacitor has resistivity 200 $$\Omega$$ m. The value of capacitance of the capacitor is 2 pF. If a potential difference of 40 V is applied across the plates of the capacitor, then the value of leakage current flowing out of the capacitor is: (given the value of relative permittivity of material is 50)

For a parallel-plate capacitor we first recall the fundamental relation for its capacitance.

Formula: $$C=\varepsilon_r\varepsilon_0\frac{A}{d}$$ where $$\varepsilon_r$$ is the relative permittivity, $$\varepsilon_0=8.854\times10^{-12}\,\text{F\,m}^{-1}$$ is the permittivity of free space, $$A$$ is the plate area and $$d$$ is the separation.

Re-arranging, the geometry factor becomes

$$\frac{A}{d}=\frac{C}{\varepsilon_r\varepsilon_0}$$.

Now we write the electrical resistance offered by the same slab of dielectric. For a material of resistivity $$\rho$$ placed between parallel plates,

Formula: $$R=\rho\frac{d}{A}$$.

We can eliminate the unknown geometry by substituting $$d/A$$ from the reciprocal of the earlier expression:

$$R=\rho\frac{d}{A}=\rho\left(\frac{1}{A/d}\right)=\rho\left(\frac{1}{C/(\varepsilon_r\varepsilon_0)}\right)=\rho\varepsilon_r\varepsilon_0\,\frac{1}{C}.$$

Substituting the numerical values $$\rho=200\,\Omega\text{ m},\; \varepsilon_r=50,\; C=2\;\text{pF}=2\times10^{-12}\,\text{F}$$, we get

$$R=200\times50\times8.854\times10^{-12}\,\frac{1}{2\times10^{-12}}$$ $$\;=10000\times8.854\times10^{-12}\,\frac{1}{2\times10^{-12}}$$ $$\;=8.854\times10^{-8}\,\frac{1}{2\times10^{-12}}$$ $$\;=\frac{8.854}{2}\times10^{(-8)-(-12)}$$ $$\;=4.427\times10^{4}\;\Omega$$ $$\;=4.427\times10^{4}\;\Omega \;(\text{about }44.3\;\text{k}\Omega).$$

The leakage current is obtained from Ohm’s law $$I=\dfrac{V}{R}$$, with the applied potential difference $$V=40\;\text{V}$$:

$$I=\frac{40}{4.427\times10^{4}}\;\text{A}$$ $$\;=0.904\times10^{-3}\;\text{A}$$ $$\;=9.04\times10^{-4}\;\text{A}$$ $$\;=0.9\;\text{mA}\;(\text{approximately}).$$

Hence, the correct answer is Option A.