How to Show the E Field Outside a Long, Charged Conducting Cylinder?

In summary, to find the magnitude of the electric field outside a long, charged conducting cylinder of radius r0 with charge density σ Cm-2, you can use the divergence theorem and consider a Gaussian surface. The surface integral of the field can be easily calculated due to symmetry, and the charge inside the Gaussian surface can be found using the surface area of the cylinder. By equating this charge to the integral of the electric field over the surface, you can solve for E. The direction of the field must also be taken into account, as well as any spatial variables that may affect its magnitude.
  • #1
Plaetean
36
0

Homework Statement


Use the divergence theorem (and sensible reasoning) to show that the E field a distance r outside a long, charged conducting cylinder of radius r0 which carries a charge density of σ Cm-2 has a magnitude E=σr0/ε0r. What is the orientation of the field?

Homework Equations



Divergence theorem

q=∫E.dAε0

The Attempt at a Solution


Completely lost really, not even sure where to start. The dimensions of σ mean that it can only be used with the surface integral (I think), and so every time I start playing around with the equations, I end up not using the divergence theorem at all. Not even sure if the second equation is relevant, just included it because I have a feeling it might be useful.
 
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  • #2
The first thing you should do is considering a Gaussian surface, which is an imaginary surface you're going to compute the surface integral on it. Then you should calculate the amount of charge you're Gaussian surface contains.The surface integral of the field is easy to handle because of the symmetry. Calculating the charge inside the Gaussian surface isn't harder because you have the surface area of the cylinder enclosed by the Gaussian surface.Try and report the results.
 
  • #3
Taking a surface integral for the sides of the cylinder I get q=2πr0σh (where h would be an arbitrary length of the cylinder). Thanks for the reply, I'm honestly not looking for someone to just answer the question for me, but I'm completely lost.
 
  • #4
That's correct.
So you have problem with the surface integral of the field?
Just think about the direction of the field and whether is depends on any spatial variables or not.Taking symmetry into account is crucial here.
 
  • #5
A cylinder of length h and radius r0 has a surface area or [itex]2\pi r_0^2h[/itex]. Since you are told that the charge density on the cylinder is [itex]\sigma[/itex], the total charge is [itex]2\pi\sigma r_0^2h[/itex]. The integral of the E field over the surface of a cylinder of radius r around that must be equal to [itex]2\pi\sigma r_0^2 h[/itex]. If E is constant, that is just E times the surface area of the cylinder or radius r.
 
  • #6
HallsofIvy said:
A cylinder of length h and radius r0 has a surface area or [itex]2\pi r_0^2h[/itex]. Since you are told that the charge density on the cylinder is [itex]\sigma[/itex], the total charge is [itex]2\pi\sigma r_0^2h[/itex].

The units of σ are Cm-2 though, not Cm-3, so you'd end up with Cm as your units for charge.
 
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FAQ: How to Show the E Field Outside a Long, Charged Conducting Cylinder?

1. What is the Divergence Theorem?

The Divergence Theorem is a mathematical theorem that relates the flow of a vector field through a closed surface to the divergence of the field inside the surface. It is also known as Gauss's Theorem or Ostrogradsky's Theorem.

2. How is the Divergence Theorem applied in real-world problems?

The Divergence Theorem is used in various fields such as physics, engineering, and fluid dynamics to calculate the flux of a vector field through a closed surface. This can be applied to problems involving fluid flow, electric and magnetic fields, and more.

3. What is the difference between the Divergence Theorem and the Divergence Operator?

The Divergence Theorem is a mathematical theorem that relates a surface integral to a volume integral, while the Divergence Operator is a mathematical operator that calculates the divergence of a vector field at a point. In simpler terms, the Divergence Theorem is a formula, while the Divergence Operator is a mathematical tool.

4. What is the significance of the Divergence Theorem in vector calculus?

The Divergence Theorem is a fundamental result in vector calculus that relates two important concepts: flux and divergence. It allows for the simplification and calculation of complex integrals and is a key tool in solving many physical and mathematical problems.

5. What are some common misconceptions about the Divergence Theorem?

One common misconception is that the Divergence Theorem only applies to 3-dimensional vector fields, when in fact it can be generalized to higher dimensions. Another misconception is that the Divergence Theorem only applies to closed surfaces, but it can also be applied to open surfaces with some adjustments. It is important to carefully consider the assumptions and limitations when using the Divergence Theorem in problem-solving.

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