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A parallel plate capacitor has plate are...

A parallel plate capacitor has plate area of A separated by distance between them. It is completely filled with dielectric medium, its dielectric constant increases linearly as move from one plate to other plate from `k_(1)` to `k_(2)` . Then the capacitance of system is given by :

A

`(3epsilon_(0)A(k_(2) - k_(1)))/(l log_(e)((k_(2))/(k_(1))))`

B

`(2epsilon_(0)A(k_(2) - k_(1)))/(l log_(e)((k_(2))/(k_(1))))`

C

`(epsilon_(0)A(k_(2) - k_(1)))/(l log_(e)((k_(2))/(k_(1))))`

D

`(epsilon_(0)A(k_(2) - k_(1)))/(2l log_(e)((k_(2))/(k_(1))))`

Text Solution

Verified by Experts

The correct Answer is:
C
`dC = (k epsilon_(0) A)/(dx)`
`k = alpha x + beta`
At x = 0, `k_(1) = beta`
At x = l, `k_(2) = alpha l + beta`
`k_(2) = alpha l + k_(1)`
`k_(2) - k_(1) = alpha l , alpha = (k_(2) - k_(1))/(l)`
`k = k_(1) + ((k_(2) - k_(1))/(l)) x`, `dc = (epsilon_(0) [(k_(1) + ((k_(2) - k_(1)))/(l))x]4)/(dx)`
`(1)/(c ) = int (1)/(dC)` , `(1)/(C ) = int_(0)^(l) (dx)/(epsilon_(0)[k_(1) + ((k_(2) - k_(1))/(l))x]4)`
`C = (epsilon_(0) A (k_(2) - k_(1)))/(l log_(e) ((k_(2))/(k_(1))))`
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