The correct option is
C q0λ2πε0log0(ba)
Charge density =λ/m
Work done to being a charge =q0
a<b
Gauss law in electrostatic, it states that total electric flux over the closed surface S is 1ϵ0 times that total charge (q) contained inside S.
∮S¯¯¯¯E0d¯¯¯s=qϵ0
Electric field due to an infinity long straight wire. Electric field due to an infinity long straight wire.
Let us consider an infinity long time charge having linear charge density λ. Assume a cylindrical Gaussian surface of radius r and length l coaxial with the line charge,
By symmetry, the electric field E has the same magnitude at each point of the curved surface S1 and is directed radially outward. So, angle at surface between ds1 and E1 is zero and angle of d¯¯¯s2, d¯¯¯s3 with ¯¯¯¯E at S2 and S3 are 900.
Total flux through the cylindrical surface,
∮¯¯¯¯E.¯¯¯ds=∮S1¯¯¯¯E.d¯¯¯s1+∮S2¯¯¯¯E.d¯¯¯s2+∮S3¯¯¯¯E.d¯¯¯s3
∮S1Eds1cos00+∮S2Eds2cos900+∮S3Eds3cos00
=E∮d¯¯¯s1=E×2πrl
Since λ is the charge per unit length and l is the length of the wire,
thus the charge enclosed q=λl
According to Gaussian law
∮¯¯¯¯E¯¯¯¯¯ds=qϵ0
or, E×2πrl=λlϵ0
E=λ2πϵ0r
Work done =−∫baqEdr
=−qλa2πϵ0log(ba)