Question on the nature of induction

[hamhamt]

Homework Statement

A long, straight wire is surrounded by a hollow metallic cylinder whose axis coincides with that of the wire. The solid wire has a charge per unit length of + λ, and the hollow cylinder has a net charge per unit length of +2λ. From this information, use Gauss's law to find:

(a) the charge per unit length on the inner and outer surfaces of the hollow cylinder

(b) the electric field outside the hollow cylinder, a distance r from the axis

Homework Equations

λ = [itex]\stackrel{Q}{A}[/itex]

The Attempt at a Solution

i'm not sure if my reasoning is correct, but this so far this is what i think based off a similar problem from the book:

since the line charge is +L, by induction, the inner surface of the cylinder is -L. consequentially, is the outer surface is the +L because of the polarization of the charge on the cylinder by the line charge?

i am not sure about how the line charge affects the outer surface of the cylinder, can anyone elaborate on this?
also, how does the polarization by the line charge affect the total charge of the system, at a distance larger than the radius of the cylinder? would this then be +L + +2L = +3L?

 

[Simon Bridge]

Just apply Gausses law.

 

[hamhamt]

i know i am supposed to apply gauss's law. the question tells you to use gauss's law.

i am unsure about what Q Enclosed is at certain regions, and is what I describe I was having issue with at the end of my post.

 

[Simon Bridge]

You are trying to do too much in advance of the calculation.

I'd try using the differential form of Gauss' law for this myself.
Pick an appropriate symmetry, and divide the volume into appropriate regions.
Solve the DE for each region - then apply the boundary conditions.
You'll have a bunch of equations which can be manipulated to help answer your questions.What does the electric field between the walls of the cylinder have to be?
If there were no line charge in the center - what distribution of charge would achieve this and why?

 

[Nickie]

I would approach this problem by first clarifying the concept of induction. Induction is the process by which an electric field can cause a redistribution of charges within a conductor, resulting in a net charge on the surface of the conductor. In this case, the long, straight wire with a charge per unit length of +λ will induce a charge on the inner and outer surfaces of the hollow cylinder, due to its electric field.

Based on Gauss's law, we can determine the charge per unit length on the inner and outer surfaces of the hollow cylinder. Since the solid wire has a charge per unit length of +λ and the hollow cylinder has a net charge per unit length of +2λ, the total charge per unit length on the inner surface of the cylinder must be -λ (as you correctly stated in your attempt at a solution) and the total charge per unit length on the outer surface must be +3λ (as the polarization of the charges on the cylinder is in the same direction as the charge on the solid wire).

To determine the electric field outside the hollow cylinder, we can use the equation λ = Q/A, where Q is the total charge on the cylinder and A is the cross-sectional area. At a distance r from the axis, the cross-sectional area is 2πr, and the total charge on the cylinder is +3λ. Therefore, the electric field at a distance r from the axis would be E = +3λ/(2πrε0), where ε0 is the permittivity of free space.

In summary, the nature of induction in this problem is the redistribution of charges on the inner and outer surfaces of the hollow cylinder due to the electric field of the solid wire. The polarization of the charges on the cylinder results in a net charge on the outer surface, which affects the total charge of the system. By using Gauss's law, we can determine the charge per unit length on the surfaces and the electric field outside the cylinder.