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Question

A capacitor is formed by two square metal-plates of edge a, separated by a distanced. Dielectric of dielectric constants K is filled in the gap as shown in the figure. The equivalent capacitance is


A

kε0a2lnkdk-1

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B

kε0a2lnkdk-2

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C

kε0a2lnk2dk-1

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D

2kε0a2lnkdk-1

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Solution

The correct option is A

kε0a2lnkdk-1


Step 1: Given data

Length of the metal edge is a and separation is d

Step 2: Solving for total capacitance:

This configuration can be considered as capacitors in series of edge dx each at distance x from one end with dielectric material increasing from one end to another, having a capacitance of dCnet.

1dCnet=d1ε0A1+d2kε0A2

Where d1 is the distance from the first end and d2 from another A1and A2 are the corresponding area, ε0 is permeability in a vacuum and k is dielectric constant

Assuming that the angle made by dielectric filling at one end is ϕ

d1=d-xtanϕ and d2=xtanϕ

where tanϕ=da

The area of there capacitance is, edge×dx=adx

1dCnet=d-xtanϕε0adx+xtanϕkε0adx

dCnet=ε0adxd-xtanϕ+xtanϕk

Integrating with respect tox with limits0toa

dCnet=ε0ka0adxkd-xktanϕ+xtanϕ

Cnet=ε0ka1-ktanϕ0-logk

Cnet=ε0ka1-kda0-logk

Cnet=ε0ka2k-1dlogk

Hence, option (A) is correct.


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