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

Options

A. zero
B. `(2 lambdaR)/(epsilon_(0))`
C. `(2lambdasqrt(R^(2)-y^(2)))/(epsilon_(0))`
D. `(lambdasqrt(R^(2)+y^(2)))/(epsilon_(0))`

Answer

c

Explanation

electric flux `oint_(s)vec(E ).vec(d)s=(q_(i n))/(epsilon_(0))`
`q_(i n)` is the charge enclosed by the Gaussian surface which, in the present case, is the surface of given sphere. As shown , length `AB` of line inside the sphere.

In `DeltaOO'A`
`R^(2)=Y^(2)+(O' A)^(2)`
`:. O'A= sqrt(R^(2)-y^(2))`
and `AB= 2sqrt(R^(2)-y^(2))`
Charge on length `AB= 2sqrt(R^(2)-y^(2))xxlambda`
`:.` Electr c flux `= oint_(s)vec(E ).vec(d)s= (2 lambda sqrt(R^(2)-y^(2)))/(epsilon_(0))`