A long non-conducting cylinder [dark gray] has a charge density = r, where = 6.5

Question

A long non-conducting cylinder [dark gray] has a charge density = r, where = 6.53 C/m^4 and r is in meters. Concentric around it is a hollow metallic cylindrical shell.

1) What is the electric field at 2 cm from the central axis? Assume the length L is very long compared to the diameter of the shell, and neglect edge effects.
Answer in units of N/C.

2)What is the electric field at 8.29 cm from the central axis?
Answer in units of N/C.

3) What is the electric field at 17.8 cm from the central axis?
Answer in units of N/C.

4) What is the surface charge density inside the hollow cylinder?
Answer in units of C/m2.

Explanation / Answer

charge density = r, = 6.53 C/m4, R = 5.16 cm

1) What is the electric field at r1 = 2 cm from the central axis? Assume the length L is very long compared to the diameter of the shell, and neglect edge effects.

use Gauss's law,

E*2r1*L = (1/0) * 2rL dr (from 0 ro r1)

E*r1 = (/0)r2 dr (from 0 ro r1)

E*r1 = (/0)r13/3

E = r12/(30) = 9.84*107 N/C.

2)What is the electric field at r2 = 8.29 cm from the central axis?

E*2r2*L = (1/0) * 2rL dr (from 0 ro R)

E*r2 = (/0)r2 dr (from 0 ro R)

E*r2 = (/0)R3/3

E = R3/(30r2) = 4.08*108 N/C.

3) What is the electric field at r3 = 17.8 cm from the central axis?

E*2r3*L = (1/0) * 2rL dr (from 0 ro R)

E*r3 = (/0)r2 dr (from 0 ro R)

E*r3 = (/0)R3/3

E = R3/(30r3) = 1.90*108 N/C.

4) What is the surface charge density inside the hollow cylinder?

surface charge inside the hollow cylinder = * 2rL dr (from 0 ro R)

= 2LR3/3

surface charge density = (2LR3/3)/(2LR) = R2/3 = 5.80*10-3 C/m2.

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