# Electric Field Equation Hollow Cylinder

• n0083
In summary, the conversation discusses the possibility of finding an equation for the electric field inside a hollow cylinder shell of finite length. It is suggested that numerical integration or series approximations may be necessary, and the details of the cylinder's properties and charge distribution are considered. Ultimately, it is concluded that finding a closed formula for the field at all points may not be possible, particularly off the axis of the cylinder.
n0083
Hello,
Does anyone know of (or have a link to) an equation for the electric field at any point inside a a hollow cylinder shell of finite length?

thanks,

I've never seen one. I suspect that it's not possible to get a closed formula for all points, and that you'd have to calculate it via numerical integration or using a series approximation of some kind.

More detail please. Is it a conducting cylinder kept at a given potential?
Are the ends open or closed?

Okay, let's simplify.

Suppose the cylinder is of finite length, and the point is inside the cylinder midway between the two ends. The ends are open. The charge is uniform and kept constant. The cylinder is made of conducting material.

That is impossible. A conducting cylinder will not have a uniform charge.

True that.

Suppose it is made of non-conducting material such that the charge is uniformly distributed.

That problem can be solved by taking the field or potential on the axis of a uniformly charged ring and integrating along the axis.

Right, on the axis you can get an equation for the field as a function of position fairly easily. Off the axis it's much more difficult, maybe even impossible, using "common" functions.

## 1. What is the electric field equation for a hollow cylinder?

The electric field equation for a hollow cylinder is given by E = λ/(2πε₀r), where λ is the linear charge density, ε₀ is the permittivity of free space, and r is the distance from the center of the cylinder.

## 2. How is the electric field inside a hollow cylinder different from the electric field outside?

Inside the hollow cylinder, the electric field is zero as there is no charge enclosed within the Gaussian surface. Outside the cylinder, the electric field is given by E = (λr)/(2πε₀R³), where R is the radius of the cylinder.

## 3. What happens to the electric field if the hollow cylinder is charged with a different charge distribution?

The electric field equation for a hollow cylinder assumes a uniform linear charge density along its length. If the charge distribution is different, the electric field will also be different and can be calculated using the appropriate equation for that charge distribution.

## 4. Can the electric field inside a hollow cylinder ever be non-zero?

No, the electric field inside a hollow cylinder will always be zero as there is no net charge enclosed within the Gaussian surface, regardless of the charge distribution on the surface of the cylinder.

## 5. How does the electric field inside a hollow cylinder change with the radius of the cylinder?

The electric field inside a hollow cylinder is inversely proportional to the radius of the cylinder. As the radius increases, the electric field decreases and vice versa.

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