2) Consider the following hollow, infinite, cylindrical, nonconducting shell of

Question

2) Consider the following hollow, infinite, cylindrical, nonconducting shell of inner radius ri and outer radius r2 with non- uniform charge distribution, +pr) por. An infinite line charge of linear density + is parallel to, and concentric with, the outer shell. Express all parts to this problem symbolically in terms of po, , r, n, r2 and fundamental constants. +p(r) Employ the following coordinate system. +2 Nonconductor Side View Cross Sectional View (a) By choosing an appropriate gaussian surface, symbolically find the electric field in the hollow space, ri>r>0 (b) By choosing an appropriate gaussian surface, symbolically find the electric field in the outer shell, r2 r>r. (c) Assume the electric potential at r rn is Vo. Symbolically find the electric potential for r. o0 e electric potential for ri >r>0 1, assuming the electr

Explanation / Answer

given, hollow, infininte non conducting holoow shell

inner radius = r1

outer radius = r2

charge distribution, rho(r) = rhoo*r

line charge density of line = lambda

a. for r < r1, r > 0

consider a concentric cylinderic hollow gaussean surface at radius r

then

from gauss law

E*2*pi*r*l = qin/epsilon

but qin = lambda*l

hence

E = lambda/2*pi*r*epsilon ( where epsilon is permittivity of free space)

b. for r2 > r > r1

charge contained insider radius r of the cylinder = Q

dQ = rho*2*pi*r*l*dr = rhoo*2*pi*r^2*l*dr

Q = integrate dQ from r = r1 to r = r

Q = rhoo*2*pi*l*(r^3 - r1^3)/3

hence just like the previous part

from gauss' law

E*2*pi*r*l = qin/epsilon

qin = rhoo*2*pi*l*(r^3 - r1^3)/3 + lambda*l

hence

E = [rhoo*2*pi*(r^3 - r1^3)/3 + lambda]/2*pi*r*epsilon

c. we know that E = -dV/dr

hence

V = - integral E dr

V = [rhoo*2*pi*(r^3/3 - r1^3*ln(r))/3 + lambda*ln(r)]/2*pi*epsilon + K ( where K is constnat of integration)

V(r = r2) = Vo

Vo = [rhoo*2*pi*(r2^3/3 - r1^3*ln(r2))/3 + lambda*ln(r2)]/2*pi*epsilon + K

K = Vo - ([rhoo*2*pi*(r2^3/3 - r1^3*ln(r2))/3 + lambda*ln(r2)]/2*pi*epsilon)

V = [rhoo*2*pi*(r^3/3 - r1^3*ln(r))/3 + lambda*ln(r)]/2*pi*epsilon + Vo - ([rhoo*2*pi*(r2^3/3 - r1^3*ln(r2))/3 + lambda*ln(r2)]/2*pi*epsilon)

d. we know that E = -dV/dr

hence

V = - integral E dr

V = lambda*ln(r)/2*pi*epsilon + K ( where K is constant of integration)

V(r1) = [rhoo*2*pi*(r1^3/3 - r1^3*ln(r1))/3 + lambda*ln(r1)]/2*pi*epsilon + Vo - ([rhoo*2*pi*(r2^3/3 - r1^3*ln(r2))/3 + lambda*ln(r2)]/2*pi*epsilon)

hence

lambda*ln(r1)/2*pi*epsilon + K = [rhoo*2*pi*(r1^3/3 - r1^3*ln(r1))/3 + lambda*ln(r1)]/2*pi*epsilon + Vo - ([rhoo*2*pi*(r2^3/3 - r1^3*ln(r2))/3 + lambda*ln(r2)]/2*pi*epsilon)

K = [rhoo*2*pi*(r1^3/3 - r1^3*ln(r1))/3 + lambda*ln(r1)]/2*pi*epsilon + Vo - ([rhoo*2*pi*(r2^3/3 - r1^3*ln(r2))/3 + lambda*ln(r2)]/2*pi*epsilon) - lambda*ln(r1)/2*pi*epsilon

hence

V(r) = lambda*ln(r)/2*pi*epsilon + [rhoo*2*pi*(r1^3/3 - r1^3*ln(r1))/3 + lambda*ln(r1)]/2*pi*epsilon + Vo - ([rhoo*2*pi*(r2^3/3 - r1^3*ln(r2))/3 + lambda*ln(r2)]/2*pi*epsilon) - lambda*ln(r1)/2*pi*epsilon

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