# Electric force inside a cylinder(my friends and I can't solve this

• catsonmars
In summary: This is a standard E-field calculation.2nd integrate z from -L/2 to L/2 to find the total force.In summary, the problem discusses a right circular cylinder with a nonuniform volume charge density that varies linearly in the z direction. The goal is to find the force on a point charge placed at the center of the cylinder. To solve this, the cylinder is sliced into disks of thickness dz and the force due to each disk is calculated using standard E-field equations. The total force is found by integrating z from -L/2 to L/2.
catsonmars
Electric force inside a cylinder(my friends and I can't solve this:(

## Homework Statement

A right circular cylinder has radius R and length L, and a nonuniform volume charge density ρ($\hat{r}$). If the z axis is chosen to coincide with the axis of the cylinder, the charge density is ρ($\hat{r}$) = ρ(z$\hat{z}$) = ρ(o)+βz This means that the charge density varies linearly along the length of the cylinder. Find the force on a point charge q placed at the center of the cylinder.

Note ρ(o) is the initial density or "ρ not". Also my picture has been cut off the line on the far right is capital L

## Homework Equations

The total force on the charge is the sum of the froces from each charge density. Note that the charge densities are not all equal. The porblem states that if were to look at the charge density as we move along the z axis it would change linearly.

F=∫dF
where dF is

The individual force equations can be written as.
dF=[κ(dq^2)/r(12)^2]$\hat{r}$(12)/r(12)
NOTE:r(12) is read as r subscript 12.

where r is the distance from the charge density to the center charge and is written as

r=(h^2+l^2)^1/2

The charge density is ρ=dq/dV
Solve for dq and substitute in ρ=ρ(o)+βz

dq=(ρ(o)+βz)(dV)

## The Attempt at a Solution

Use these equations to solve for dF to find the total force. This is where I start getting confused. First before I begin. The problem makes it sound as if the charge density is exclusively on the z axis, is that correct. If that is the case I am just integrating the density from 0 to L. This feels wrong though. Also this would make my picture wrong! Here is what I have so far.dF=[κ(ρ(o)+βz)(dV)]$\hat{r(12)}$/r^3

substitute in r=(h^2+l^2)^1/2

dF=[κ(ρ(o)+βz)(dV)]$\hat{r(12)}$/(h^2+l^2)^3/2

Now integrate both sides.

∫dF=κ∫[(ρ(o)+βz)(dV)]$\hat{r(12)}$/(h^2+l^2)^3/2

I am integrating over F in the first integral and in the second one I am integrating from 0 to L.
If my picture is right I feel like I am missing two more integrals so I could integrate over the volume. Where am I going wrong or what am I missing?

catsonmars said:
First before I begin. The problem makes it sound as if the charge density is exclusively on the z axis, is that correct.
No. The problem states that the charge density varies only in the z direction - implies it is uniform in the x-y directions.

You need to slice the cylinder into disks thickness dz.
1st find the expression for the force due to the disk of charge at position z.

## 1. What is electric force inside a cylinder?

Electric force inside a cylinder refers to the force exerted by electric charges on each other within a cylindrical space. This force is caused by the interaction between charged particles, such as electrons, and is a fundamental force in physics.

## 2. How is electric force calculated inside a cylinder?

The magnitude of electric force inside a cylinder can be calculated using Coulomb's law, which states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. The direction of the force is determined by the charges' polarity.

## 3. What factors affect the electric force inside a cylinder?

The electric force inside a cylinder is affected by the amount of charge present, the distance between the charges, and the properties of the material surrounding the charges. The presence of other charged particles or external electric fields can also influence the force.

## 4. How can I solve for the electric force inside a cylinder?

To solve for the electric force inside a cylinder, you will need to know the charges present, the distance between them, and the properties of the surrounding material. You can then use Coulomb's law to calculate the force, or you can use mathematical equations specific to cylindrical geometries.

## 5. Can you provide an example of calculating electric force inside a cylinder?

Sure, let's say we have two point charges of +2 microcoulombs and -3 microcoulombs, separated by a distance of 5 centimeters inside a non-conductive cylinder. Using Coulomb's law, we can calculate the force between them to be 1.8 x 10^-7 newtons. Alternatively, we can use the equation Fe = (kQ1Q2)/r^2, where k is the Coulomb's constant, Q1 and Q2 are the charges, and r is the distance between them, to arrive at the same result.

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