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NEET Class neet Answered

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Asked by Prashant DIGHE | 27 Jan, 2020, 10:42: PM
answered-by-expert Expert Answer
Let us divide the capacitor plate that makes small angle α with the line parallel to other plate,
into small lengths of size dl as shown in figure.
 
Then the capacitor becomes parallel combination of many number of capacitors having plates of width dl .
 
Let us consider one such small capacitor of capacitance dC at a distance l from one end of plate.
 
capacitance dC = εo (a × dl) / (d+ l tanα )
 
where a is the side of square plate and d is the initial distance between plates.
 
capacitance C of all combined capacitors is given by

begin mathsize 14px style C space equals space epsilon subscript o a space integral subscript 0 superscript a fraction numerator d l over denominator d space plus space l space tan alpha end fraction space equals space fraction numerator epsilon subscript o a over denominator tan alpha end fraction space open square brackets ln space open parentheses d space plus space l space tan alpha close parentheses close square brackets subscript 0 superscript a space equals space fraction numerator epsilon subscript o a over denominator tan alpha end fraction space ln fraction numerator open parentheses d space plus space a space tan alpha close parentheses over denominator d end fraction end style.........................(1)
For small angle α, tanα ≈ α .  Logarithmic term can be written as
 
begin mathsize 14px style ln fraction numerator d plus space a space tan alpha over denominator d end fraction space equals space ln open square brackets 1 space plus space open parentheses fraction numerator a space alpha over denominator d end fraction close parentheses close square brackets space almost equal to space open parentheses fraction numerator a space alpha over denominator d end fraction close parentheses space minus space 1 half space open parentheses fraction numerator a space alpha over denominator d end fraction close parentheses squared end style..............................(2)
[ when x << 1, ln(1+x) = x - ( x2 / 2 ) + higher order of x . Higher orders are neglected in eqn.(2)  ]

Hence , using eqn.(2), eqn.(1) is written as
 
begin mathsize 14px style C space equals space fraction numerator epsilon subscript o space a over denominator alpha end fraction open square brackets fraction numerator a space alpha over denominator d end fraction space minus space 1 half space fraction numerator a squared space alpha squared over denominator d squared end fraction close square brackets space equals space space fraction numerator epsilon subscript o space a squared over denominator d end fraction open square brackets 1 space minus space fraction numerator a space alpha over denominator 2 d end fraction close square brackets end style
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