What is the electric field inside an infinite cylinder?

[Jaccobtw]

Homework Statement

**Disclaimer** I tried to write a more descriptive title but it wouldn't let me write enough characters.

An infinitely long, hollow cylinder of radius 5cm has a charge of 30nC/m. Note that the number I just gave you is not the surface charge density. It is the charge per unit length, but you can use it to get the surface charge density if you want. Since the cylinder is infinite, there are no end caps to consider. What is the Electric Field in N/C at a point 3cm from the axis down the center of the cylinder?

Relevant Equations

Gauss' Law

3 cm is inside the cylinder. We can use a gaussian cylinder to enclose the inside of the cylinder up to 3 cm. Because the outer cylinder is infinite there is no flux out of the end caps with the inner cylinder. There is also no charge enclosed in the cylinder. So the electric field 3cm away from inside the enclosed cylinder is zero. Is this correct reasoning or am I missing something here? Thank you

 

[SammyS]

Homework Statement:: **Disclaimer** I tried to write a more descriptive title but it wouldn't let me write enough characters.

An infinitely long, hollow cylinder of radius 5cm has a charge of 30nC/m. Note that the number I just gave you is not the surface charge density. It is the charge per unit length, but you can use it to get the surface charge density if you want. Since the cylinder is infinite, there are no end caps to consider. What is the Electric Field in N/C at a point 3cm from the axis down the center of the cylinder?
Relevant Equations:: Gauss' Law

3 cm is inside the cylinder. We can use a gaussian cylinder to enclose the inside of the cylinder up to 3 cm. Because the outer cylinder is infinite there is no flux out of the end caps with the inner cylinder. There is also no charge enclosed in the cylinder. So the electric field 3cm away from inside the enclosed cylinder is zero. Is this correct reasoning or am I missing something here? Thank you

Your reasoning is not complete enough to state that the electric field 3cm away from inside the enclosed cylinder is zero.

You have only given enough reason to state that the flux through the closed cylinder is zero.

You need to include some argument(s) appealing to symmetry.

 

[Jaccobtw]

Your reasoning is not complete enough to state that the electric field 3cm away from inside the enclosed cylinder is zero.

You have only given enough reason to state that the flux through the closed cylinder is zero.

You need to include some argument(s) appealing to symmetry.

$$\oint_{}^{} E \cdot dA = \frac{q_e}{\epsilon_o}$$

We know know that the electric field only points out of the long face of the cylinder and not the two end caps so we will do Gauss's law just for that face.

The field vectors are normal to the surface so:
$$\oint_{}^{} \left| E \right|\left| dA \right| = \frac{q_e}{\epsilon_o}$$

The electric field is constant everywhere on the surface so we can move E outside the integral

$$\left| E \right|\oint_{}^{} \left| dA \right| = \frac{q_e}{\epsilon_o}$$

The integral of dA is A

$$\left| E \right| \left| A \right| = \frac{q_e}{\epsilon_o}$$

Divide both sides by A

$$\left| E \right| = \frac{q_e}{\left| A \right|\epsilon_o}$$

The charge enclosed is zero:

$$\left| E \right| = \frac{0}{\left| A \right|\epsilon_o}$$

Therefore, E = 0

 

[Jaccobtw]

Your reasoning is not complete enough to state that the electric field 3cm away from inside the enclosed cylinder is zero.

You have only given enough reason to state that the flux through the closed cylinder is zero.

You need to include some argument(s) appealing to symmetry.

How’s the above argument?

 

[kuruman]

Homework Statement:: **Disclaimer** I tried to write a more descriptive title but it wouldn't let me write enough characters.

An infinitely long, hollow cylinder of radius 5cm has a charge of 30nC/m. Note that the number I just gave you is not the surface charge density. It is the charge per unit length, but you can use it to get the surface charge density if you want. Since the cylinder is infinite, there are no end caps to consider. What is the Electric Field in N/C at a point 3cm from the axis down the center of the cylinder?
Relevant Equations:: Gauss' Law

3 cm is inside the cylinder. We can use a gaussian cylinder to enclose the inside of the cylinder up to 3 cm. Because the outer cylinder is infinite there is no flux out of the end caps with the inner cylinder. There is also no charge enclosed in the cylinder. So the electric field 3cm away from inside the enclosed cylinder is zero. Is this correct reasoning or am I missing something here? Thank you

Strictly looking at the statement of the problem, how do you know that there is no charge enclosed by a gaussian cylinder of radius 3 cm? You are not told that the charge per unit length is all on the surface and you are not told that the cylinder is conducting. The hollow cylinder could be non-conducting with a volume charge distribution such that the charge per unit length would be 30 nC/cm. What argument(s) can you think of to support your assertion that there is zero charge enclosed? I don't disagree with you, I am only asking how you can justify your assertion.

 

[Jaccobtw]

Strictly looking at the statement of the problem, how do you know that there is no charge enclosed by a gaussian cylinder of radius 3 cm? You are not told that the charge per unit length is all on the surface and you are not told that the cylinder is conducting. The hollow cylinder could be non-conducting with a volume charge distribution such that the charge per unit length would be 30 nC/cm. What argument(s) can you think of to support your assertion that there is zero charge enclosed? I don't disagree with you, I am only asking how you can justify your assertion.

It says the cylinder is hollow. So I assumed the charge can only be on the surface

 

[SammyS]

It says the cylinder is hollow. So I assumed the charge can only be on the surface

I agree.
I read the problem as to mean that the cylinder has a rather thin wall of uniform thickness with a uniform charge density . Yes, some big assumptions on my part, but otherwise Gauss's Law won't do you much good.

Beyond that, you need to argue from symmetry that the electric field has particular characteristics of cylindrical symmetry. In other threads of yours, @Orodruin has given you specific conditions under which Gauss's Law can be used to determine electric field values.

 

[kuruman]

It says the cylinder is hollow. So I assumed the charge can only be on the surface

Only if the cylinder is conducting or if the problem says that all the charge is on the surface. There is no mention of either of these conditions. Furthermore, since there could be charge inside the cylinder, there could or could not be charge at 3 cm depending on how thick the wall of the hollow cylinder is, a number that is not given. Too many could be's and unknowns hanging in mid air for my taste.

My point is that the statement of the problem is too vague for one to give an answer without making the assumption that all the charge is on the surface. You need to be aware that it is an assumption and not a conclusion based on the statement of the problem. If you want a rigorous justification of your solution, you must add this assumption to what you already have.