An infinitely long line charge having a uniform charge per unit length `lambda` lies at a distance d from a point O as shown in figure. Determine the total electric flux through through the surface of a sphere of radius R centered at O resulting from this line charge. Consider both cases where `Rltd` and `Rgtd`.

Options

A. zero
B. `(2lambdaR)/(epsilon_0)`
C. `(2lambdasqrt(R^2-y^2))/epsilon_0`
D. `(lambdasqrt(R^2+y^2))/epsilon_0`

Answer

C

Explanation

c. Electric flux `oint_S vecE * vecd s = q_(in)/epsilon_0`
`q_(in)` is the charge enclosed by the Gaussian surface, which in
the present case, is the surface of the given sphere. As shown,
length AB of line lies 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 s `2 sqrt(R^2 - Y^2) xx lambda`
Therefore, electric flux is
`oint_S vecE * vecd s = (2lambda sqrt(R^2 -y^2))/epsilon_0`