A capacitor is made of two square plates each of side ‘a’ making a very small angle α between them, as shown in figure. The capacitance will be close to :

Question

A capacitor is made of two square plates each of side ‘a’ making a very small angle  α between them, as shown in figure. The capacitance will be close to :

Infinity Learn · Accepted Answer

⇒Select in small strip of dx, at a distance x from left.​⇒$$tan$$α$$=yx$$⇒$$y=xtan$$α$$=x$$αFor small angles $$tan$$α$$=$$α​⇒distance between plates at a distance $$x=d+x$$α⇒Capacitance of this element can be written as, ​⇒$$dC=$$ε$$0dAd+$$αx⇒$$dC=$$ε$$0adxd+$$αx      ​⇒All such elements are connected in parallel, integration will give total capacitance ⇒∫$$0adC=$$∫$$0a$$ε$$0adxd+$$αx ⇒$$C=$$ ε$$0a$$∫$$0adxd+$$αx​⇒$$C=$$ε$$0a$$α$$lnd+$$α$$x0a$$ ∵ ∫$$1axdx=logaxddxax+C$$ ⇒$$C=$$ε$$0a$$α$$ln1+$$αad​⇒C≃ε$$0a2d1$$−α$$a2d$$            $$ln1+x$$ $$=x1$$−$$x2$$

A capacitor is made of two square plates each of side ‘a’ making a very small angle $α$ between them, as shown in figure. The capacitance will be close to :

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