-) A very long, nonconducting cylindrical shell has an inner radius A, an outer

Question

-) A very long, nonconducting cylindrical shell has an inner radius A, an outer radius B, and a nonuniform charge density given by (r)-+ re-r where and are constants. The nonconducting spherical shell is surrounded by a concentric, nonconducting cylindrical shell with inner radius B, outer radius C and uniform charge density such that the magnitude of the electric field for r> C is zero. a. b. c. d. What is the total charge contained in the first cylindrical shell? What is the charge density in the second cylindrical shell? What is the magnitude of the electric field for A

Explanation / Answer

given a very long , non conducting cylindrical shell, inner radius A, outer radius B

charge density rho(r) = 3*alpha/r^2 + r*e^(-beta*r)

another non conducting shell, inner radius B, outer radius C

uniform charge density rho

E r > C = 0

a. total charge in the first shell = Q

dQ = 2*pi*r*dr*l*rho(r)

dQ = 2*pi*l*r*[3*alpha/r^2 + r*e^(-beta*r)]dr [ l is length of the cyulinder]

integrating form r = A to r = B

Q = 2*pi*l[3*alpha*ln(B/A) + e^(-beta*B)[-2/beta^3 - 2B/beta^2 - B^2/beta] - e^(-beta*A)[-2/beta^3 - 2A/beta^2 - A^2/beta]]

b. charge density in second shell = rho

now, E for r > C = 0

hence form gasuss law

charge on outer shell + charge on inner shell = 0

hence

rho = -Q/pi(C^2 - B^2)l = -2[3*alpha*ln(B/A) + e^(-beta*B)[-2/beta^3 - 2B/beta^2 - B^2/beta] - e^(-beta*A)[-2/beta^3 - 2A/beta^2 - A^2/beta]]/(C^2 - B^2)

c. for A < r < B

from gauss' law

E*2*pi*r*l = 2*pi*l[3*alpha*ln(r/A) + e^(-beta*r)[-2/beta^3 - 2r/beta^2 - r^2/beta] - e^(-beta*A)[-2/beta^3 - 2A/beta^2 - A^2/beta]]/epsilon

E = [3*alpha*ln(r/A) + e^(-beta*r)[-2/beta^3 - 2r/beta^2 - r^2/beta] - e^(-beta*A)[-2/beta^3 - 2A/beta^2 - A^2/beta]]/epsilon*r

d. for B < r < C

fomr gauss' law

E*2*pi*r*l = [2*pi*l[3*alpha*ln(r/A) + e^(-beta*r)[-2/beta^3 - 2r/beta^2 - r^2/beta] - e^(-beta*A)[-2/beta^3 - 2A/beta^2 - A^2/beta]] - 2pi*l(r^2 - B^2)*l[3*alpha*ln(B/A) + e^(-beta*B)[-2/beta^3 - 2B/beta^2 - B^2/beta] - e^(-beta*A)[-2/beta^3 - 2A/beta^2 - A^2/beta]]/(C^2 - B^2)]/epsilon

E = [[3*alpha*ln(r/A) + e^(-beta*r)[-2/beta^3 - 2r/beta^2 - r^2/beta] - e^(-beta*A)[-2/beta^3 - 2A/beta^2 - A^2/beta]] - (r^2 - B^2)*l[3*alpha*ln(B/A) + e^(-beta*B)[-2/beta^3 - 2B/beta^2 - B^2/beta] - e^(-beta*A)[-2/beta^3 - 2A/beta^2 - A^2/beta]]/(C^2 - B^2)]/epsilon*r

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