Finding the Electric Field of a Hollow Cylindrical Conductor

In summary, the conversation discusses how to find the electric field of a hollow, open-ended, thin-walled cylindrical conductor. It is mentioned that the net electric field within the cylinder is ostensibly zero, but there may be a net electric field near the open boundaries of the cylinder. Several methods, including numerical solutions and integration, are suggested to solve this problem. Ultimately, it is concluded that the net electric field within a finite hollow cylinder is never zero except at the midpoint of the axis.
  • #1
tade
702
24
I am trying to find the electric field of a hollow, open-ended, thin-walled cylindrical conductor.

I am trying to solve something regarding LINACs:

300px-Lineaer_accelerator_en.svg.png


Assume we have an isolated, hollow, open-ended, thin-walled cylindrical conductor, with a net charge.

The net electric field within the cylinder is ostensibly zero, but since it isn't a closed surface, is there a net electric field near the open boundaries of the cylinder?

I tried to solve it using integral methods, but I ended up having to integrate over a function involving elliptic integrals of the first kind, which is hard, and I'm stupid. :confused:
 
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  • #2
Cool problem. If you're interesting in a numerical solution why not do it numerically. Use some integration like trapezoidal integration over the charges.
 
  • #3
tade said:
The net electric field within the cylinder is ostensibly zero
no it can never be zero for a finite hollow cylinder,(except on points lying on mid plane (normal to axis) of cylinder which i doubt may also be nonzero but at midpoint of axis)
suppose take a point P on axis of cylinder at a distance 't' from one base of cylinder (t < (h/2)) where h is height of cylinder
field at P due to portion of cylinder below P (ie of height 't') has some non zero projection along axis = E1
now take portion of cylinder above P, divide it into 2 parts , 1st part which is symmetric(about P) to portion of cylinder below P, rest part is 2nd part
field at P due to 1st part (ie of height 't') has some non zero projection along axis which cancels E1 (since vertical component of force is in opposite dir)
field at P due to 2nd part (ie of height 'h-2t') has some non zero projection along axis
so there is net non-zero vertical projection of field at any point on axis inside finite hollow cylinder (except at midpoint of axis)
so resultant field strength vector must be nonzero in magnitude for any point inside finite hollow cylinder
(except on points lying on mid plane (normal to axis) of cylinder which i doubt may also be nonzero but at midpoint of axis)
 

1. How do you determine the electric field of a hollow cylindrical conductor?

To determine the electric field of a hollow cylindrical conductor, you can use Gauss's Law, which states that the electric flux through a closed surface is equal to the charge enclosed by that surface divided by the permittivity of free space. This can be expressed mathematically as E = Q/(2πε_0r), where E is the electric field, Q is the charge enclosed, ε_0 is the permittivity of free space, and r is the distance from the center of the cylinder.

2. What is the significance of a hollow cylindrical conductor in determining the electric field?

A hollow cylindrical conductor is significant because the electric field inside a conductor is zero, so the only contribution to the electric field comes from the charge on the surface of the conductor. In a hollow cylinder, all of the charge is located on the outer surface, making it easier to calculate the electric field using Gauss's Law.

3. Can the electric field of a hollow cylindrical conductor be uniform?

Yes, under certain conditions, the electric field of a hollow cylindrical conductor can be considered uniform. This is true when the length of the cylinder is much greater than its radius, and when the distance from the center of the cylinder is much greater than its radius. In this case, the electric field can be approximated as constant throughout the cylinder.

4. How does the electric field inside a hollow cylindrical conductor vary with distance from the center?

The electric field inside a hollow cylindrical conductor is zero, regardless of the distance from the center. This is because the charge is located on the outer surface of the cylinder, so there is no electric field inside the conductor. However, as you move away from the cylinder, the electric field increases according to the inverse square law.

5. What are some real-world applications of finding the electric field of a hollow cylindrical conductor?

One important application is in the design of electrical circuits and equipment. Understanding the electric field of a hollow cylindrical conductor can help engineers and scientists determine the most efficient and effective placement of conductors in a circuit. This knowledge is also used in the development of medical imaging techniques, such as magnetic resonance imaging (MRI), which use cylindrical conductors to create a uniform magnetic field. Additionally, the electric field of a hollow cylindrical conductor is important in the study of electromagnetic waves and their propagation through different materials.

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