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Find the equation of the equipotentials for an infinite cylinder of radius r_(0), carrying charge of linear density lambda. |
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Answer» Solution :Fig SHOWS an infinite cyclinder of radius `r_(0)` carrying charge of lineardensity `lambda`, From symetry,we find that the fieldlines MUST be radially outwards. Imaginea cylindrical Gaussian surface of radius r and length L. According to Gauss's theroem, `ointvec(E), vec(DS) = (q)/(in_(0)) = (lambda l)/(in_(0))` `E(2 pi r l) = (lambda l)/(in_(0)) or E = (lambda)/(2pi in_(0) r)` ..(i) `:. V (r) - V(r_(0)) = int_(r_(0))^(r) vec(E). vec(dl) = (lamda)/(2pi in_(0)) log_(e) (r_(0))/(r)` For an equipotential surface of GIVEN V(r), `log_(e) (r)/(r_(0)) = (2pi in_(0))/(lambda) [V (r) - V (r_(0))] :. r = r_(0) e^(-2 pi in_(0) [V(r) - V(r_(0))]//lambda)`...(ii) Hence, equipotential surfaces are cylinders of radius r given by (ii). 
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