Deriving Electric Field Inside Plastic Rod Using Gauss's Law

In summary, the conversation discusses finding the electric field inside a straight circular plastic cylinder that is irradiated with a beam of protons and has a total excess charge Q. The discussion involves using Gauss's Law and a Gaussian surface in the shape of a sphere to find the electric field, but it is pointed out that this may not be the most efficient method due to the symmetry of the problem. The conversation then delves into the symmetry of the problem and how it can be used to simplify the problem and find the electric field using cylindrical coordinates. The conversation also mentions performing a triple integral and a surface integral to evaluate the flux, and it is noted that these calculations are not complicated due to the constant charge density.
  • #1
yeezyseason3
16
0

Homework Statement



A straight circular plastic cylinder of length L and radius R (where
R ≪ L)
is irradiated with a beam of protons so that there is a total excess charge Q distributed uniformly throughout the cylinder. Find the electric field inside the cylinder, a distance r from the center of the cylinder far from the ends, where r < R.

Homework Equations


Gauss's Law where ∫Edl = q/e0 where q is the charge inside a certain area. Volume of sphere: 4/3πr^3 and volume of a cylinder: 2πr^2L with r being radius of cylinder and L being the length of the cylinder.

The Attempt at a Solution


I decided to make my gaussian surface a sphere which was located inside the cylinder. The charge inside that sphere would then be Q(Asphere/Acylinder). I then divided that by e0. I set that equal to the electric field multiplied by the area of the sphere. Like it is supposed to - the surface area of the spheres cancel leaving me with E = Q/(e0*Acylinder). The surface area of the cylinder is given as 2piRL + 2piR^2 so my final answer is Q/(e0*(2piRL + 2piR^2)) but apparently that is not correct. I have a feeling I messed up my charge distribution somehow because I believe that I have to find the charge inside the sphere as a function of volume instead of surface area.
 
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  • #2
What is your argument for using a sphere as a gaussian surface ? Does that make use of any symmetry, and of r << R in any way ?
 
  • #3
BvU said:
What is your argument for using a sphere as a gaussian surface ? Does that make use of any symmetry, and of r << R in any way ?
I am working under the assumption that since charge is evenly distributed throughout the cylinder, that a gaussian sphere would act like a point charge.
 
  • #4
Maybe, but there's more charge in the neighborhood, and that charge also contributes to the electric field !
The gauss theorem is always true, but it is most useful if you can work it around in such a way that the E field is perpendicular to the surface everywhere, so that ##\vec E \cdot d\vec A## becomes ##|\vec E| dA##. For that you need to exploit some symmetry present. The wording of your exercise very clearly points at the symmetry (symmetries) you need to exploit here !
 
  • #5
BvU said:
Maybe, but there's more charge in the neighborhood, and that charge also contributes to the electric field !
The gauss theorem is always true, but it is most useful if you can work it around in such a way that the E field is perpendicular to the surface everywhere, so that ##\vec E \cdot d\vec A## becomes ##|\vec E| dA##. For that you need to exploit some symmetry present. The wording of your exercise very clearly points at the symmetry (symmetries) you need to exploit here !
How am I supposed to find symmetry if I do not know the pattern of the field? I assume that it is evenly distributed, but I cannot draw any other conclusions.
 
  • #6
You can assume the cylinder is infinitely long (take it from me). You can pick the origin of a coordinate system anywhere on the axis of the cylinder. Describing features of a cylinder is easiest with cylindrical coordinates . Does it matter where we pick z = 0 ? No. That's a symmetry. Does it matter whch way is the z+ ? No. Thats another symmetry. And so on with ##\phi##. Not with r (sometimes designated ##\rho##), but we chose r = 0 on the axis (for good reasons).
 
  • #7
BvU said:
You can assume the cylinder is infinitely long (take it from me). You can pick the origin of a coordinate system anywhere on the axis of the cylinder. Describing features of a cylinder is easiest with cylindrical coordinates . Does it matter where we pick z = 0 ? No. That's a symmetry. Does it matter whch way is the z+ ? No. Thats another symmetry. And so on with ##\phi##. Not with r (sometimes designated ##\rho##), but we chose r = 0 on the axis (for good reasons).

Isn't cylindrical coordinates, with phi, theta and rho? What are you referring to as z+? Even with that information how are you supposed to perform a triple integral to evaluate the flux? If so, will the integrand just be 1 or is there a particular function to integrate? If there is, would it be the cylinder represented in cylindrical coordinates?
 
  • #8
The blue letters indicate a clickable link. There you can find that cylindrical coordinates describe 3d space with a radius, a z and a phi. Here we use r for the radius because he symbol ##\rho## is used for charge density.

The origin can be chosen anywhere on the axis of the cylinder -- you get the same situation. (Translation symmetry). That means ##|\vec E (r,\phi,z)|## can not depend on ##z##. But there still could be a z component.

The positive z axis has to point in some direction, e.g. up. The mirror symmetry of this problem is that if you let it point down, then you get the same problem. That means the E field has no z component: if it had, there would be two solutions and there is only one solution for the E field (the solution of the Laplace equation is unique). So far ##\vec E(r,\phi,z) = (E_r(r,\phi),E_\phi(r,\phi), 0)##.

Digest, research, check etc. these (in physics very important) steps and try to reduce ##\vec E(r,\phi,z) = (E_r(r,\phi),E_\phi(r,\phi), 0)## to ##\vec E(r,\phi,z) = (E_r(r),0, 0)## yourself.
yeezyseason3 said:
Even with that information how are you supposed to perform a triple integral to evaluate the flux? If so, will the integrand just be 1 or is there a particular function to integrate? If there is, would it be the cylinder represented in cylindrical coordinates?
A triple integral of the constant charge density ##\rho## isn't that complicated.
And the surface integral isn't complicated either if you can work things around in such a way that the E field is perpendicular to the surface everywhere (see #4) :smile: .

Don't your notes or textbook include examples of using gauss law ?

--
 

What is Gauss's Law?

Gauss's Law is a fundamental law of electromagnetism that describes the relationship between electric charges and the electric field they create. It states that the electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space.

How can Gauss's Law be used to derive the electric field inside a plastic rod?

Gauss's Law can be applied to any closed surface, including a cylindrical surface around the plastic rod. By using a Gaussian surface and considering the charge enclosed by that surface, the electric field inside the rod can be calculated.

Why is a plastic rod used in this derivation instead of a metal rod?

Plastic is an insulating material, meaning it does not allow electric charges to flow through it easily. This allows for a more simplified derivation, as there are no free charges inside the plastic rod that would contribute to the electric field.

What are the assumptions made when using Gauss's Law to derive the electric field inside a plastic rod?

The main assumptions made in this derivation are that the plastic rod is a perfect cylinder with uniform charge distribution and that the electric field inside the rod is also uniform. Additionally, the electric field outside the rod is assumed to be zero.

How is this derivation relevant in real-world applications?

Understanding the electric field inside a plastic rod is important in various applications, such as designing electrical insulation materials or analyzing the behavior of charged particles in plastic-based devices. This derivation provides a theoretical framework for understanding and predicting the behavior of electric fields in plastic materials.

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