# Electric Field Integration Outside a Cylinder

• frostmephit
In summary, the problem asks to find the charge on an alpha particle placed 5.0 cm away from the end of a thin, hollow cylinder with a charge of -90nC. The cylinder has a radius of 0.5 cm and length of 2.0 cm. Using the Electric Field of a Ring equation and dividing the cylinder into infinitesimally small rings, an integral can be set up to calculate the charge on each ring. This will then give the total charge on the cylinder, which can then be used to find the electric field at the location of the alpha particle.
frostmephit

## Homework Statement

A thin, hollow cylinder missing its two end caps is shown to have a charge of -90nC. It has a radius of .5 cm and is 2.0 cm long. Find the charge on an $$\alpha$$ particle 5.0cm away from the end of the cylinder closest to the particle. (The center of the cylinder is on the same axis as the alpha particle.)

## Homework Equations

(Electric Field of a Ring)
E=1/(4$$\pi\epsilon$$$$\o$$) * q$$\Delta$$z/(R^2+$$\Delta$$z^2), where R is the radius of the cylinder and z is the distance from the center of the ring to the observation location, q is the charge, and the charge of an alpha particle is given by 2e

## The Attempt at a Solution

The way I have gone about doing this is in such a manner as to divide the cylinder up into so many small rings, each with an infinitesimaly small charge. The problem is, I can't figure out how to create an equation representing the whole cylinder. I am aware that the charge on the rings would be equal to the surface area of one of them divided by the surface area of the greater cylinder times the charge, but I've been having trouble representing this as an equation for integration. Am I going in the right direction, and could someone please help?

Welcome to Physics Forums.
frostmephit said:

## Homework Equations

(Electric Field of a Ring)
E=1/(4$$\pi\epsilon$$$$\o$$) * q$$\Delta$$z/(R^2+$$\Delta$$z^2), where R is the radius of the cylinder and z is the distance from the center of the ring to the observation location, q is the charge, and the charge of an alpha particle is given by 2e

## The Attempt at a Solution

The way I have gone about doing this is in such a manner as to divide the cylinder up into so many small rings, each with an infinitesimaly small charge. The problem is, I can't figure out how to create an equation representing the whole cylinder. I am aware that the charge on the rings would be equal to the surface area of one of them divided by the surface area of the greater cylinder times the charge, but I've been having trouble representing this as an equation for integration. Am I going in the right direction, and could someone please help?
Okay. First, I'll recommend using z instead of Δz, for the distance from the alpha particle to the ring.

If each ring has a "length" of dz along the z-direction, what would be the charge q on that ring? You're expression for q should contain a "dz" in it, so that will (hopefully) suggest an integral.

## 1. What is the formula for calculating the electric field outside a cylinder?

The formula for calculating the electric field outside a cylinder is E = (Q * r) / (2 * π * ε0 * L), where E is the electric field, Q is the charge of the cylinder, r is the distance from the cylinder, ε0 is the permittivity of free space, and L is the length of the cylinder.

## 2. How does the electric field outside a cylinder differ from inside a cylinder?

The electric field inside a cylinder is uniform and has a constant magnitude, while the electric field outside a cylinder decreases with distance from the cylinder. Additionally, the direction of the electric field inside a cylinder is always perpendicular to the surface of the cylinder, while outside the cylinder it can vary in direction.

## 3. What factors can affect the electric field outside a cylinder?

The electric field outside a cylinder can be affected by the charge of the cylinder, the distance from the cylinder, the permittivity of the surrounding medium, and the length of the cylinder.

## 4. How is electric field integration used to calculate the electric field outside a cylinder?

Electric field integration is used by dividing the cylinder into small, infinitesimal charge elements and integrating the electric field contribution of each element over the entire surface of the cylinder. This allows for the calculation of the total electric field outside the cylinder.

## 5. What are some real-world applications of the electric field outside a cylinder?

The electric field outside a cylinder is important in understanding the behavior of electrical equipment, such as capacitors and transmission lines. It is also used in the design of electric motors, generators, and other electromagnetic devices. Additionally, it is essential in the study of atmospheric electricity and lightning strikes.

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