An infinite cylinder of radius R has a linear charge density . The volume charge
ID: 1326217 • Letter: A
Question
An infinite cylinder of radius R has a linear charge density . The volume charge density (C/m3)within the cylinder (rR) is (r)=0rR, where 0 is a constant to be determined.
A.
The charge within a small volume dV is dq=dV. The integral of dV over a cylinder of length L is the total charge Q=L within the cylinder. Use this fact to to determine the constant 0 in terms of and R.
Hint: Let dV be a cylindrical shell of length L, radius r, and thickness dr. What is the volume of such a shell?
Express your answer in terms of the variables , R, and appropriate constants.
B.
Use Gauss's law to find an expression for the electric field E inside the cylinder, rR. Give your answer as a multiple of /0.
Express your answer in terms of the variables r, R, and appropriate constants.
Explanation / Answer
Gauss’s law give a field strength of E = enclosed=20r,
where enclosed = 0 r dV is the charge within a unit length of coaxial cylindrical surface of radius r,
and dV = 2 r dr is the volume element for a unit length of thin shell with this surface.
(a) For r < R (inside cylinder), enclosed = 0 r (2 0=R)r 2 dr = 20r 3 =3R, hence E = 0r 2 =30R.
(b) For r > R (outside cylinder), enclosed = 0 R(20=R)r 2 dr = 20R2 =3 , hence E = 0R2 =30r.