A parallel plate capacitor has plates of area $A$ separated by distance $d$ between them. It is filled with a dielectric which has a dielectric constant that varies as $\mathrm{k}(\mathrm{x})=\mathrm{K}(1+\alpha \mathrm{x})$ where $\mathrm{x}$ is the distance measured from one of the plates. If $(\alpha \text {d)}<<1,$ the total capacitance of the system is best given by the expression
A parallel plate capacitor has plates of area $A$ separated by distance $d$ between them. It is filled with a dielectric which has a dielectric constant that varies as $\mathrm{k}(\mathrm{x})=\mathrm{K}(1+\alpha \mathrm{x})$ where $\mathrm{x}$ is the distance measured from one of the plates. If $(\alpha \text {d)}<<1,$ the total capacitance of the system is best given by the expression
- [JEE MAIN 2020]
- A
$\frac{\mathrm{AK} \varepsilon_{0}}{\mathrm{d}}\left(1+\frac{\alpha \mathrm{d}}{2}\right)$
- B
$\frac{\mathrm{A} \varepsilon_{0} \mathrm{K}}{\mathrm{d}}\left(1+\left(\frac{\alpha \mathrm{d}}{2}\right)^{2}\right)$
- C
$\frac{\mathrm{A} \varepsilon_{0} \mathrm{K}}{\mathrm{d}}\left(1+\frac{\alpha^{2} \mathrm{d}^{2}}{2}\right) $
- D
$ \frac{\mathrm{AK} \varepsilon_{0}}{\mathrm{d}}(1+\alpha \mathrm{d})$
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