Question

Can someone pleas explain how to do this problem as thoroughly possible. Thank you  

2. An infinite cylinder lies on the z-axis with the following charge density: ρ(s) Po, 0, sa where s is the cylindrical radial coordinate. γ and Po are constants. Deter- mine the electric field in the regions s-b, b < s < a, and s-a. For s-a, express your answer in terms of an appropriately defined charge per unit length

Answer 1

P(s) = gamma/s e^-s/b s lessthanorequalto b = P_0 b < s < a = 0 s greaterthanorequalto a s greaterthanorequalto a Consider a co axial cylindrical surface S of length L and radius S. (s greaterthanorequalto a) By symmetry E^bar Vertical-bar Vertical-bar ds^bar Vertical-bar Vertical-bar s^^on the curved part of S and E^bar perpendicular ds^bar on flat ends of S Gauss law countourintegral_s E^bar ds^bar = countourintegral_s E ds = E countourintegral_s ds = (E)(2 pi sl) Q_enc = integral edv = integral^a_0 (2 pi s l ds) e = integral^b_0 gamma/s e^-s/b 2 pi sl ds + integral^a_b (2 pi sds)(P_0 l) = 2 pi gamma l integral^b_0 e^-s/b ds + (2 pi P_0 l) integral^a_0 sds = (2 pi gamma l) e^-s/b/-1/b Vertical-bar ^b_0 + (2 pi P_0 l) s^2/2 Vertical-bar ^a_b = (2 pi gamma l b) (1 - e^-1) + (pi P_0 l)(a^2 - b^2) \r\n charge per unit length lambda = Q/l lambda = (2 pi b gamma)(1 - e^-1) + pi