# Electric field inside a conducting cylinder with wire

• Liquidxlax
In summary, a Geiger counter is a device used to detect ionizing radiation. It consists of a central wire surrounded by a hollow cylindrical conductor, with a high voltage applied to the wire to create a strong electric field inside the cylinder. When radiation enters the cylinder, it ionizes gas atoms and creates a chain reaction of free electrons, which are collected by the wire and generate a signal. In order to calculate the amount and polarity of charge on the central wire, the equations for electric field and charge distribution on a cylinder can be used. Gauss' law can also be applied to determine the electric field contribution of a charged, hollow cylinder inside the cylinder.
Liquidxlax

## Homework Statement

You don't really need to read the top paragraph.

A Geiger counter is used to detect ionizing radiation. The detector consists of a thin wire that is surrounded by a concentric circular conducting cylinder. A high voltage is applied to the wire so that it has a positive charge and the surrounding cylinder has the same amount of negative charge. This establishes a very strong electric field inside the cylinder. A low pressure inert gas is inside the cylinder, so that when radiation enters the cylinder it ionizes a few of the gas atoms, and the resulting free electrons are attracted to the positive wire. The e field is so strong as to enable the gas atoms between collisions to gain enough energy to ionize these atoms as well, creating more free electrons, and a chain reaction ensues. An "avalanche" of electrons reaches the wire that is collected by the wire and generates a signal.

Suppose the radius of the central wire is 25x10^(-6)m, the cylinder has a 0.014m radius and the cylinder length is 0.16m. The electric field magnitude at the cylinder wall is 2.9x10^(4) N/C. What is the amount of charge and it's polarity on the central wire?

## Homework Equations

Er = (4pikQ)/(2pirL) cylinder charge distribution

rho = Q/L

surface area cylinder 2piRL

## The Attempt at a Solution

E = kQ/2piRL + KQ/2pirL

can't get the latex or w.e to work hold on

Hello Liquidxlax,
Liquidxlax said:

## Homework Equations

Er = (4pikQ)/(2pirL) cylinder charge distribution
Okay, that equation is good for the electric field in the region outside of a very long, charged wire or cylinder. Which is good, because we are concerned with a region outside of a central wire for now.

If you didn't derive the equation yourself, you can use Gauss' law to derive it, if you wanted to. :

$$\oint_S \vec E \cdot d \vec A = \frac{Q_{enc}}{\epsilon_0}$$

$$E(2 \pi r \ell) = \frac{Q_{enc}}{\epsilon_0}$$

$$E = \frac{Q_{enc}}{(2 \pi r \ell)\epsilon_0}$$

And noting that $k = 1/(4 \pi \epsilon_0)$

$$E = \frac{4 \pi k Q_{enc}}{2 \pi r \ell}$$

[Edit: And by the way, r here is the radius of the hypothetical Guassian surface, not the radius of the wire itself. It is assumed that r is greater than the radius of the wire, since we're trying to find the electric field contribution from the wire, in the region outside of the wire.]

The reason why I went through all of that was to demonstrate Gauss' law. I'd like you to use it again below.
rho = Q/L
So far so good.
surface area cylinder 2piRL
For our purposes, yes, that's right (we can ignore the end-caps).

## The Attempt at a Solution

E = kQ/2piRL + KQ/2pirL

Umm, I'm not sure where that came from either.

It seems to me like you are trying to find the electric field contribution of the charged, hollow cylinder, and incorrectly applying it inside the cylinder. But you're not doing something right.

You might be able to easily figure it out by finding the electric field contribution of a charged, hollow cylinder, inside the cylinder.

You can use Gauss' law to derive that too. Forget about the wire in the center for the moment (we've already found the electric field contribution of that). Use Gauss' law to find the electric field inside of a thin, hollow, charged cylinder.

$$\oint_S \vec E \cdot d \vec A = \frac{Q_{enc}}{\epsilon_0}$$

I'll let you take it from here (hint: the charge enclosed within the region inside of a thin, hollow, charged cylinder is zero -- there isn't any charge on the inside, it's all on the cylinder's surface! ).

Last edited:
lol well if that is true, what is the significance of the cylindrical shell around the, but not to signify a radius at which an electric field was measured?

ps. my prof sucks and sucked at trying to teach gauss's law. So my understanding of it isn't very goodoh and I'm not sure about the polarity thing still

Liquidxlax said:
lol well if that is true, what is the significance of the cylindrical shell around the, but not to signify a radius at which an electric field was measured?
For regions inside the thin, hollow, long cylinder, the charge on the cylinder has no bearing on the electric field inside.

This is the same idea for a hollow, spherical shell. The electric field inside of a hollow spherical shell (with uniform charge distribution about its surface) is always zero, as long as there are no other charges around.

So what is the significance of the cylinder at all for this particular problem? Well, ask yourself what is the electric field in the region outside of the cylinder (with the oppositely charged wire inside). Of course, before you get there, your first question is what is the total charge enclosed within the Gaussian surface outside of the cylinder. But now there are two things that are charged: the wire and the cylinder.
ps. my prof sucks and sucked at trying to teach gauss's law. So my understanding of it isn't very good
Gauss' law is a great tool, but it takes practice.
oh and I'm not sure about the polarity thing still
The problem statement says that the wire has a positive charge, and the cylinder has a negative charge of the same magnitude. In other words, the charge on the wire is Q, and the charge on the cylinder is -Q.

collinsmark said:
Gauss' law is a great tool, but it takes practice.

yep but he started it yesterday and we have a midterm on tuesday... and 3 assigments due that tuesday and another midterm the next day. So practice is limitedbut thanks for the help, i think I'm understanding it better

## 1. What is an electric field inside a conducting cylinder with wire?

An electric field inside a conducting cylinder with wire refers to the distribution of electric charges and the resulting force within a cylindrical object that is made of a material that allows the flow of electricity. The presence of a wire within the cylinder can affect the electric field by altering the distribution of charges.

## 2. How is the electric field inside a conducting cylinder with wire calculated?

The electric field inside a conducting cylinder with wire can be calculated using the formula E = σ/ε, where E is the electric field, σ is the surface charge density, and ε is the permittivity of the material. The presence of a wire can also change the electric field by introducing an additional source of charges.

## 3. Why is the electric field inside a conducting cylinder with wire important?

The electric field inside a conducting cylinder with wire is important because it can help us understand the behavior of electricity within a cylindrical object. It also has practical applications in electrical engineering, such as in the design of circuits and electronic devices.

## 4. How does the electric field inside a conducting cylinder with wire differ from that of a solid cylinder?

The electric field inside a conducting cylinder with wire differs from that of a solid cylinder because of the presence of the wire. The wire acts as an additional source of charges, which can alter the distribution of the electric field within the cylinder. This can lead to different patterns and strengths of the electric field.

## 5. Can the electric field inside a conducting cylinder with wire be manipulated?

Yes, the electric field inside a conducting cylinder with wire can be manipulated by changing the properties of the wire, such as its length, thickness, and material. The presence of other objects or materials near the cylinder can also affect the electric field. In addition, by controlling the flow of electricity through the wire, the electric field within the cylinder can be altered.

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