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A capacitor has rectangular plates of le...

A capacitor has rectangular plates of length a and width b. The top plate is inclined at a small angle as shown in. The plate separation varies from `d=y_(0)` at the left to `d=2y_(0)` at the right, where `y_(0)` is much less than a or b. system.
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Text Solution

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We consider a differential strip of width `dx` and length b to approximate a differential capacitor of area `bdx` and separation

`d=y_(0)+(y_(0)/a)x`.
All such differential capcitors are in parallel arrangement. Thus,
`dC=(epsilon_(0)(bdx))/(y_(0)+(y_(0)/a)x)` or `C=intdC`
or `C=epsilon_(0)bint_(0)^(a)(dx)/(y_(0)+y_(0)/ax)`
`=(epsilon_(0)b)/((y_(*0)//a))[In((y_0+y_(0)/axxa)/y_(0))]=(epsilon_(0)ab)/y_(0) In 2`
We can determine the expression for capacity in terms of `theta` as
`d =(y_(0)+x tan theta)`
`C =intdC =int_(b)^(a)(epsilon_(0)b dx)/(y_(0)+x tan theta)`
`=(epsilon_(0))/(tan theta)` In `(y_(0)+tan theta)/y_(0)`
For small `theta`,
`tan theta=theta` or `C=(epsilon_(0)b)/theta In(1+(atheta)/y_(0))`
Now, we can use the expansion
`log (1+x)=x-1/2x^(2)+`...
For `xlt1`, we can negleet higher powers. Thus,
`C=(epsilon_(0)b)/(theta)[(atheta)/y_(0)-1/2((atheta)/y_(0))^(2)]=(epsilon_(0)ab)/y_(0)[1-(atheta)/(2y_(0))]`
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