Question

Derive the expression for the capacitance of a parallel plate capacitore having plate area A and plate separation d.

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Solution
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Let a parallel plate capacitor be applied a voltage across its terminals due to which a charge  develops across its ends. 
Separation between plates is and area of plates is .
From gauss's theorem one can show electric field inside the capaciton plates ,
, where (charge / Area)
Now, voltage across the plates is related by :
[In general , but we take as is uniform]
Now, capacitance is defined by , thus we get :-

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