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Find the equatio of the equipotentials f...

Find the equatio of the equipotentials for an infinite cylinder of radius `r_(0)` carrying charge of linear density `lamda`.

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Fig shows an infinite cyclinder of radius `r_(0)` carrying charge of linear density `lambda`, From symetry, we find that the field lines must be radially outwards. Imagine a cylindrical Gaussian surface of radius r and length l. According to Gauss's theroem,
`oint vec(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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