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Find the equation of the equipotential f...

Find the equation of the equipotential for an infinite cylinder of radius `r_(0)` carrying charge of linear density `lambda`.

Text Solution

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Take a Gaussian surface of radius r and length l

`int_(0)^(2pil)vecE.dvecS= (Q)/(in_(0))`
`= (lambdal)/(in_(0))`
From Gauss.s law
`[ E_(r) S cos theta]_(0)^(2pirl) = (lambdal)/(in_(0))`
`E_(r)xx2pirl=(lambdal)/(in_(0))[ theta=0 :. cos 0^(@)= 1]`
`:. E_(r)= (lambda)/(2pi in_(0)r)`
The radius of infinite cylinder is `r_(0)`
`V(r)- V(r_(0))=-int_(r_(0))^(r)Edl`
`= -(lambda)/(2piin_(0))"log"_(e)(r)/(r_(0))=(lambda)/(2piin_(0))="log"_(e)(r_(0))/(r)`
because `int_(r_(0))^(r)(lambda)/(2piin_(0))dr= (lambda)/(2p in_(0))int_(r_(0))^(r)(1)/(r) dr`
`V = (lambda)/(2pi in_(0))"log"_(e)(r)/(r_(0))`
For given V ,
`"log"_(e)(r)/(r_(0))=-(2piin_(0))/(lambda)xx[V(r)-V(r_(0))]r=r_(0)e ^((2pi in_(0))/(lambda)[V(r)-V(r_(0))])`
`:. r = r_(0)e^(-(2pi in_(0))/(lambda)[V(r)-V(r_(0))])`
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