A long copper cylindrical shell of inner radius 5 cm and outer radius 8 cm surrounds concentrically a charged long aluminum rod of radius 1 cm with a charge density of 7 pC/m. All charges on the aluminum rod reside at its surface. The inner surface of the copper shell has exactly opposite charge to that of the aluminum rod while the outer surface of the copper shell has the same charge as the aluminum rod. Find the magnitude and direction of the electric field at points that are at the following distances from the center of the aluminum rod:

(a) 0.5 cm,
(b) 1.5 cm,
(c) 2.5 cm,
(d) 3.5 cm,
(e) 7 cm.

a. at distance 0.5cm from the center of the rod, this point falls on the rod itself and since the charge spread out on the surface of the rod, there wont be any electric field inside the rod itself

Hence E=0 at 0.5cm

b. at 1.5cm i.e 0.015m

the electric field is expressed as

[tex]E=\frac{2*charge density}{4\pi E_{0}r}\\E=\frac{2*7*10^{-12} *9*10^{9}}{0.015}\\E=8.4N/C[/tex]

The direction of the field depends on the charge on the rod

c. at 2.5cm i.e 0.025m

the electric field is expressed as

[tex]E=\frac{2*charge density}{4\pi E_{0}r}\\E=\frac{2*7*10^{-12} *9*10^{9}}{0.025}\\E=5.04N/C[/tex]

The direction of the field depends on the charge on the rod

d. at 3.5cm i.e 0.035m this point is still within the rod and the inner cylinder

the electric field is expressed as

[tex]E=\frac{2*charge density}{4\pi E_{0}r}\\E=\frac{2*7*10^{-12} *9*10^{9}}{0.035}\\E=3.6N/C[/tex]

The direction of the field depends on the charge on the rod

e. at 7cm which is a point outside the rod and the cylinder, the electric field is

[tex]E=\frac{2*charge density}{4\pi E_{0}r}\\E=\frac{2*7*10^{-12} *9*10^{9}}{0.07}\\E=1.8N/C[/tex]

The direction of the field depends on the charge on the rod