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A uniformly charged and infinitely long ...

A uniformly charged and infinitely long line having a linear charge density `lambda` is placed at a normal distance y from a point O. Consider a sphere of radius R with O as centre and Rgty. Electric flux through the surface of the sphere is

A

zero

B

`(2lambdaR)/(epsilon_(0))`

C

`(2lambda sqrt(R^(2)-y^(2)))/(epsilon_(0))`

D

`(lambda sqrt(R^(2)+y^(2)))/(epsilon_(0))`

Text Solution

Verified by Experts

The correct Answer is:
C
Electric flux `oint_(s)vecE.vec(dS)=(q_("in"))/(epsilon_(0)) q_("in")` is the charge enclosed by the Gaussian-surface which, in the present case, is the surface of given sphere. As shown, length AB of the line lies inside the sphere.

In `DeltaO O.A" "R^(2)=y^(2) +(O.A)^(2)`
`:. O.A= sqrt(R^(2)-y^(2))`
and `AB=2 sqrt(R^(2)-y^(2))`
Charge on length `AB=2 sqrt(R^(2)-y^(2)) xx lambda`
`:.` electric flux `=oint_(s)vecE.vec(dS)=(2lambda sqrt(R^(2)-y^(2)))/(epsilon_(0))`
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