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Question

Figure shows a parallel plate capacitor having square plates of edge length a and separation d. The gap between the plates is filled with a dielectric of dielectric constant K which varies parallel to an edge as, K=Ko+αx
Where, K and α are constants and x is the distance from the left end. Calculate the capacitance of system.


A
ϵoaKoα2d
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B
ϵoa2Kod
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C
ϵoa2(Ko+aα)2
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D
ϵoa2d(Ko+aα2)
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Solution

The correct option is D ϵoa2d(Ko+aα2)

Consider a small strip of width dx at a separation x from the left end.

This small strip forms a small capacitor of plate area

dA=adx

The capacitance of small capacitor,

dC=(Ko+αx)ϵo(adx)d

[C=KAϵod]

Arrangement of such small strips starting from x=0 to x=a will give rise to the entire capacitor.

And, since all small capacitors will be connected in parallel ( because their end terminals are connected at same points i.e., A & B)

Therefore,

Cnet=dC

Cnet=a0(Ko+αx)ϵoadxd

Cnet=Koϵoad[x]ao+αaϵod[x22]ao

Cnet=Koϵoad[a0]+αaϵo2d[a20]

Cnet=ϵoa2d[Ko+aα2]

Therefore, right option is (b)
Why this question?
Tip: It intends to challenge the understanding of combination of capacitors. If we look carefully we can infer that all small capacitors are connected in parallel, hence integration will be used to obtain summation of all capacitance.
or,Cnet=dC

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