Gausss Law of length L has an insulating cylinder of radius

Gauss\'s Law of length L has an insulating cylinder of radius a at its center and an outer insulating ring of inner radius b and radius c on the outer ring is given throughout the cylindrical ring. The volume bis as Qo and is given p ra) between radius a and density p within the inner cylinder is space. arbitrary r sa (a) Total charge Q within an radius (b) Total charge enclosed within the radius r a for (d) field for b srsb (e) Electric a sa (f field for r field at r 0 (g) Electric field at r b cylindrical (h) field at r c rdrdod: in are only (ij field at (b) Recall dV we the ends. G) Electric coordinates for (a) (c) (f). Since cylindrical of length you may neglect coordinates. Use Gaussian cylinder radially outward the electric field at points Show all your work.

Solution

(A) taking a thin shell of radius r and width dr.

Volume, dV = 2 pi r L dr

charge on this part, dq = rho0 (r / a^2) (2 pi r L) dr

dq = (2 pi rho0 L / a^2 ) ( r^2 dr)

integrating,

q = (2 pi rho0 L / a^2 ) (r^3 / 3)

r is from 0 to r

q = (2 pi rho0 L / a^2 ) (r^3 / 3)

(B)

r is from 0 to a.

q = (2 pi rho0 L / a^2 ) (a^3 / 3) = 2 pi rho0 a L / 3

(C) for r >= c

total charge inclosed, Qin = q + Qo

from Gauss law,

total flux = E. A = Qin / e0

E (2 pi r L ) = [ 2 pi rho0 a L / 3 + Qo] / e0

E = (rho0 a / 3 e0 r) + ( Q0 / 2 pi e0 r L )

(D) b < r < c

Qin = 2 pi rho0 a L / 3 + (Qo (r^2 - b^2) / (c^2 - b^2))

E (2 pi r L) = [2 pi rho0 a L / 3 + (Qo (r^2 - b^2) / (c^2 - b^2)) ] / e0

E = (rho0 a / 3 e0 r) + (Q0 (r^2 - b^2) / 2 pi e0 r L (c^2 - b^2))

(e) a< r < b

Qin = 2 pi rho0 a L / 3

E (2 pi r L) = [2 pi rho0 a L / 3 ] / e0

E = rho0 a / 3 e0 r

(F) r < a

Qin = (2 pi rho0 L / a^2 ) (r^3 / 3)

E ( 2 pi r L ) = (2 pi rho0 L / a^2 ) (r^3 / 3) / e0

E = (rho0 r^2 ) / (3 e0 a^2)

(g) r = 0

E = 0

(h) r = a

E = rho0 a / 3 e0 r

putting r = a

E = rho0 / 3e0

(i) r= b

E = rho0 a / 3 e0 r

putting r = b

E = rho0 a / 3 e0 b

(j) E = (rho0 a / 3 e0 r) + (Q0 (r^2 - b^2) / 2 pi e0 r L (c^2 - b^2))

putting r = c

E = (rho0 a / 3 e0 c) + (Qo (c^2 - b^2) / 2 pi e0 c L (c^2 - b^2))

E = (rho0 a / 3 e0 c) + (Qp / 2 pi e0 c L )