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An infinite cylinder of radius r(o), car...

An infinite cylinder of radius `r_(o),` carrying linear charge density `lamda.` The equation of the equipotential surface for the cylinder is

A

`r=r_(0)e^(pi epsi_(0)[V(r)+V(r_(0))]lamda)`

B

`r=r_(0)e^(2piepsi_(0)[V(r)-V(r_(0))]lamda^(2))`

C

`r=r_(0)e^(-2piepsi_(0)[V(r)=V(r_(0))]lamda)`

D

`r=r_(0)e^(-2piepsi_(0)[V(r)-V(r_(0))]lamda)`

Text Solution

Verified by Experts

The correct Answer is:
C
Gaussian surface of radius r and length l.
According to Gauss's theorem
`oint vecE . vecds = (q)/(epsi_(0)) = (lambdal)/(epsi_(0))`
` E (2 pi l) = (lambdal)/(epsi_(0)) or E = (lambda)/(2piepsi_(0)r)" "....(i)`
`therefore V(r) - V(r_(0)) = int_(r_(0))^(1) vecE vec(dl)=(lambda)/(2 pi epsi_(0)) log_(e). (r_(0))/(r)`
For an equipotenital surface of given V(r).
`log _(e) .(r)/(r_(0)) = (2piepsi_(0))/(lambda) [ V (r) - V(r_(0))]`
`therefore r = r_(0) e^(-2piepsi_(0)[V(r) - V(r_(0))]lambda)`
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