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A capacitor has two square plates each o...

A capacitor has two square plates each of sidel making an angle `theta` between them as shown in Fig. Calculate capacitor of the arrangement for small values of `theta`

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Let us consider an elementary strip of width dx at a distance x from left end of the capacitor. The capacitance of this element is

`dc = (in_(0) (l dx))/(r )` …(i)
From the Fig.
`r = d + x tan theta = d + x theta`, when `theta` is small,
`= d (1+ (x theta)/(d))`
From (i),
`dC = (in_(0) l dx)/(d (1+ (x theta)/(d))) = (in_(0) l)/(d) (1+ (x theta)/(d))^(-1) dx`
`dC = (in_(0) l)/(d) (1- (x theta)/(d)) dx`
`:. C = (in_(0) l)/(d) [ int_(0)^(l) dx - int_(0)^(l) (theta)/(d) x dx]`
`= (in_(0) l)/(d) [ t - (theta)/(d) (l^(2))/(2)]`
`C = (in_(0) l^(2))/(d) [l - (theta l)/(2d)]`
This is the required expression for capacity of the given condenser.
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