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 $$kx$$$$=K01+$$αx where ‘x’ is the distance measured from one of the plates. If αd

Question

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  $$kx$$$$=K01+$$αx where ‘x’ is the distance measured from one of the plates. If  αd<<$$1$$, the total capacitance of the system is best given by the expression

Infinity Learn · Accepted Answer

Capacitance of “dx” element  C'$$=K$$ε$$0Adx$$ Capacitance of “dx” element  C'$$=K01+$$αxε$$0AdxAll$$ such elements are connected in series. Hence, net capacitance is given by, ∑$$1C$$'$$=$$∫$$0ddxK0$$ε$$0A1+$$αx               ⇒$$1C=1K0$$ε$$0A$$α$$ln1+$$α$$x0b$$∵∫$$1axdx=logaxa+c$$ ⇒$$1C=1K0$$ε$$0A$$α$$ln1+$$αd     Given    αd<<$$1$$    ⇒$$1C=1K0$$ε$$0A$$ααd−α$$2d22$$      [using $$ln(1+x)$$ $$=$$ $$x-x22+$$...]⇒$$1C=dK0$$ε$$0A1$$−α$$d2$$                            ⇒$$C=K0$$ε$$0Ad1-$$α$$d2-1$$ ⇒$$C=K0$$ε$$0Ad1+$$α$$d2$$         [using    $$(1+x)n$$  $$=$$ $$1+nx$$  ]

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