An infinitely long uniform line charge distribution of charge per unit length λ lies parallel to the $$y-axis$$ in the $$y-z$$ plane at $$z=32a$$ (see$$ $$figure). If the magnitude of the flux of the electric field through the rectangular surface ABCD lying in the $$x-y$$ plane with its centre at the origin is $$z=$$λLnε$$0$$ $$($$ ε$$0$$ $$=permittivity$$ of free $$space)$$, then the value of n is

Question

An infinitely long uniform line charge distribution of charge per unit length λ lies parallel to the $$y-axis$$ in the $$y-z$$ plane at $$z=32a$$  (see$$ $$figure). If the magnitude of the flux of the electric field through the rectangular surface ABCD lying in the $$x-y$$ plane with its centre at the origin is  $$z=$$λLnε$$0$$ $$($$ ε$$0$$ $$=permittivity$$ of free $$space)$$, then the value of n  is

Infinity Learn · Accepted Answer

Imagine , a cylindrical gaussian surface of length L and radius a.$$tan$$∠$$APO=a23a2=13$$​⇒∠$$APO=300$$​Now , ∠$$APD=2$$∠$$APO=600Given$$ rectangular surface ABCD subscribes an angle  $$600$$ at the infinity long uniform line of charge.The total angle subscribe by cylinder is $$3600$$ and charge enclosed is λL.Flux through the rectangle is same as the flux through $$16th$$ of the hexagon since it subtends $$600$$ at P and field lines are symmetrically radially outwards.ϕ$$cyl=q$$∈$$0$$​∴ϕ$$ABCD=q6$$∈$$0=$$λ$$L6$$∈$$0$$​⇒$$n=6$$

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