A long non-conducting cylinder [dark gray] has a charge density p = alpha r, whe

Question

A long non-conducting cylinder [dark gray] has a charge density p = alpha r, where alpha = 4.21 C/m1 and r is in meters. Concentric around it is a hollow metallic cylindrical shell (light gray in the figure below).- What is the direction of the electric field at 14.4 cm from the central axis? The electric field is not a vector and therefore has no direction. Undetermined, since the field is zero. Points radially inward. Points radially outward. What is the electric field at 19.2 cm from the central axis? Answer in units of N/C What is the surface charge density inside the hollow cylinder? Answer in units of C/m2

Explanation / Answer

the location of 14.4 cm is inside the conducting cylinder. Thus, the field is zero. So answer is 2.

(019) The point given here lies outside both cylinders. The situation is cylinderically symmetric. Thus the field depends only on the total charge per unit length along the axis of the cylinder inside.

The total charge per unit length is thus sum of charges per unit length of the outer and inner surfaces of outer cylinder and the inner cylinder.

Since, the outer cylinder is overall electrically neutral, thus the charge per unit length on its inner surface should cancel the charge per unit length on outer surface. Thus, total charge per unit length is same as the charge per unit length on the inner cylinder.

Now, we evaluated the charge per unit length on inner cylinder here ( http://www.cramster.com/answers-may-12/advanced-physics/physics_2559619.aspx?rec=0 )

we got, (lambda=7.822*10^{-4}C/m)

thus, field at r = 19.2 cm = 0.192 m is (E=rac{lambda}{2pi epsilon_0 r}=rac{7.822*10^{-4}}{2pi*8.854*10^{-12}*0.192}=7.64*10^7 N/C)

(020) Since, the electric field inside the hollow cylider is zero, thus if we take a gauss surface to be a cylinder having the same axis as that of hollow and inner cylinders and having a radius such that its surface lies inside the outer cylinder. Then, total electric flux going out through it is zero as the electric field is zero on its surface.

Thus net charge per unit length enclosed within this gauss surface is zero.

Thus, net charge per unit length on the inner surface of hollow cylinder is negative of the charge per unit length on the inner cylinder. (lambda_{2}=-lambda=-7.822*10^{-4}C/m)

this is charge of a cylinderical part of the inner surface of length 1 m. Also, the area of inner surface having length 1 m is (A = 2pi r*1)

where r is the radius of inner surface of outer cylinder. (A=2pi*0.106*1=0.67 m^2)

thus, surface charge density is (sigma=-7.822*10^{-4}/0.67=1.167*10^{-3}C/m^2)

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