Find the electric field at an arbitrary position

[BPBAIR]

Problem:

Use Gauss’s Law to find the electric field at an arbitrary position r in the hollow region (ρ =0) between two concentric conducting cylinders. The inner cylinder has radius a and bears a constant charge per unit length, λ, while the outer cylinder has inner radius b and bears the same charge per unit length.

Equation:

Solution:
Not really sure, where to go with this from a geometric point of view.

 

[siddharth]

Ok, first of all, what symmetry can you use in this problem? Is it spherical, cylindrical, etc?

Also, just think of the definition of Gauss' Law, and see where you can apply it here.

 

[BPBAIR]

Well it would have cylindrical symettry. And the field would be completely radial b/c the top and bottom components would cancel leaving only the integral for the sides. rho=0 for in between the two cylinders, which resembles a coaxial cable with a space between the two cylinders. If anyone could help me furtehr on this problem, I would appreciate it. I am confused on how to setup the integrand since it is between two cylinders.

 

[siddharth]

Well it would have cylindrical symettry. And the field would be completely radial b/c the top and bottom components would cancel leaving only the integral for the sides. rho=0 for in between the two cylinders, which resembles a coaxial cable with a space between the two cylinders. If anyone could help me furtehr on this problem, I would appreciate it. I am confused on how to setup the integrand since it is between two cylinders.

Ok. First of all, you need to construct a hypothetical gaussian surface over which you need to evaluate the surface integral, which in this case will be a cylinder because of the symmetry. First, can you find out what is the elemental area you're integrating over? Can you proceed from here to find the field?

 

[BPBAIR]

figured it out. only the inner cylinder will contribute to the field. from there it is two simple intergrals and the gaussian turns out yo be [ lambda / (2*pi*epsilon*r) ] = E