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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

`(2lambdasqrt(R^2-y^2))/(epsilon_0)`

D

`(lambdasqrt(R^2+y^2))/(epsilon_0)`

Text Solution

Verified by Experts

The correct Answer is:
C

Electric flux `oint_(S)ver(E).vec(d)S=(q_("in"))/(epsilon_(0))`
`q_("in")` is the charge enclosed by the gaussian surface which is the present case, is the surface of given sphere. As shown length AB of the line inside the sphere in `triangleOO'A`
`R^(2)=y^(2)+(O'A)^(2)`
or `O'A=sqrt(R^(2)-y^(2))` and `AB=sqrt(R^(2)-y^(2))`
charge on length `AB=2sqrt(R^(2)-y^(2))xxlamda`
Electric flux `oint_(S)ver(E).vec(d)S=(2lamdasqrt(R^(2)-y^(2)))/(epsilon_(0))`
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