NEET Class neet Answered
The gap between the plates
![question image](http://images.topperlearning.com/topper/new-ate/top_mob1657478472134579074376785ce0-25cf-4a1f-a0f9-d13d555865d6.jpg)
Asked by padamkumar52010 | 11 Jul, 2022, 00:11: AM
Potential difference between plates that are separated by a distance d is given as
![begin mathsize 14px style V space equals space integral subscript 0 superscript d E space d y end style](https://images.topperlearning.com/topper/tinymce/cache/875aad146d163bf3b5765088a88b61b1.png)
where E is elctric field at a distance y
![begin mathsize 14px style E space equals space fraction numerator sigma over denominator space epsilon left parenthesis y right parenthesis space epsilon subscript o end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/e29acf127fe633eda8fca07d10973795.png)
where σ is charge density on plates and ε(y) is variable dielectric constant and εo is permittivity of free space
ε(y) = ε1 + m y , where m = ( ε2 - ε1 ) /d
Hence potential difference is obtained from eqn.(1) as
![begin mathsize 14px style V space equals space sigma over epsilon subscript o integral subscript 0 superscript d fraction numerator d y over denominator epsilon subscript 1 plus m y end fraction space equals space fraction numerator sigma over denominator m space epsilon subscript o end fraction open square brackets ln left parenthesis epsilon subscript 1 plus m y right parenthesis close square brackets subscript 0 superscript d space end style](https://images.topperlearning.com/topper/tinymce/cache/1435c23b8f912a59fc6774b1e409e7cf.png)
![begin mathsize 14px style V space equals space fraction numerator sigma over denominator m space epsilon subscript o end fraction space ln space open parentheses epsilon subscript 2 over epsilon subscript 1 close parentheses space space equals space fraction numerator sigma space d over denominator left parenthesis epsilon subscript 2 minus epsilon subscript 1 right parenthesis space epsilon subscript o end fraction space ln space stretchy left parenthesis epsilon subscript 2 over epsilon subscript 1 stretchy right parenthesis space end style](https://images.topperlearning.com/topper/tinymce/cache/a6a9016141ad9d1bb150950d2654808c.png)
charge density σ = Q / A , where Q is charge on plates and A is area of plate .
Hence above expression for potential difference becomes
![begin mathsize 14px style V space equals Q over A space cross times fraction numerator space d over denominator space left parenthesis epsilon subscript 2 minus epsilon subscript 1 right parenthesis space epsilon subscript o end fraction space ln space stretchy left parenthesis epsilon subscript 2 over epsilon subscript 1 stretchy right parenthesis space end style](https://images.topperlearning.com/topper/tinymce/cache/09c14b83260e6dd43a8969c8131d37d5.png)
Capacitance C = Q / V
![begin mathsize 14px style C space equals space fraction numerator open parentheses epsilon subscript 2 minus epsilon subscript 1 close parentheses over denominator ln open parentheses begin display style epsilon subscript 2 over epsilon subscript 1 end style close parentheses end fraction space cross times fraction numerator epsilon subscript o A over denominator d end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/b42c7569e59c100b2672442910ccd823.png)
Answered by Thiyagarajan K | 11 Jul, 2022, 21:08: PM
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