Electrostatics Question 265
Question: nnCapacitance of a capacitor made by a thin metal foil is $ 2\mu F $ . If the foil is folded with paper of thickness$ 0.15mm $ , dielectric constant of paper is 2.5 and width of paper is$ 400mm $ , then length of foil will be [RPET 1997]
Options:
A) $ -\frac{Q}{4}(1+2\sqrt{2}) $
B) $ \frac{Q}{4}(1+2\sqrt{2}) $
C) $ -\frac{Q}{2}(1+2\sqrt{2}) $
D) $ \frac{Q}{2}(1+2\sqrt{2}) $
Show Answer
Answer:
Correct Answer: B
Solution:
If all charges are in equilibrium, system is also in equilibrium.
Charge at centre: charge q is in equilibrium because no net force acting on it corner charge:
If we consider the charge at corner B. This charge will experience following forces $ F _{A}=k\frac{Q^{2}}{a^{2}}, $
$ F _{C}=\frac{kQ^{2}}{a^{2}} $ ,$ F _{D}=\frac{kQ^{2}}{{{(a\sqrt{2})}^{2}}}\text{and}F _{O}=\frac{KQq}{{{(a\sqrt{2})}^{2}}} $
Force at B away from the centre = $ F _{AC}+F _{D} $
$ =\sqrt{F _{A}^{2}+F _{C}^{2}}+F _{D}=\sqrt{2}\frac{kQ^{2}}{a^{2}}+\frac{kQ^{2}}{2a^{2}}=\frac{kQ^{2}}{a^{2}}( \sqrt{2}+\frac{1}{2} ) $
Force at B towards the centre $ =F _{O}=\frac{2kQq}{a^{2}} $
For equilibrium of charge at B, $ F _{AC}+F _{D}=F _{O} $
therefore $ \frac{KQ^{2}}{a^{2}}( \sqrt{2}+\frac{1}{2} )=\frac{2KQq}{a^{2}} $
therefore $ q=\frac{Q}{4}( 1+2\sqrt{2} ) $
