Electric Field On Charged Cylinder

In summary, to find the electric field at all values of radius for an infinitely long cylinder of charge, you can use Gauss's law and cylindrical coordinates. The volume charge density is given by \rho = \rho_{0}(\frac{r}{R}) for r<R and the integral form of Gauss's law can be used to find Q_{enc}. The left hand side of Gauss's law can be simplified by using a cylindrical Gaussian surface, and after equating the two sides and solving for E(r), you can find the electric field for r<R. For r>R, further consideration is needed for Q_{enc}.
  • #1
Lancelot59
646
1
I need to find the electric field at all values of radius for an infinitely long cylinder of charge. It's in insulator of radius R, and has a volume charge density
[tex]\rho = \rho_{0}(\frac{r}{R})[/tex] while r<R.

I need to find the electric field at all points first off. I'm not entirely sure how to go about doing this. I have gauss's law, but I'm not sure how to go about applying it in this case.
 
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  • #2
I would start with the integral form of Gauss's law.
[tex] \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0} [/tex]

In this case your [itex] Q_{enc} [/itex] will be a function of r, and can be found using the volume integral in cylindrical coordinates

[tex]Q_{enc}= \int \rho d\tau = \int \rho r dr d\theta dz[/tex]

and integrating the radius from 0 to r (since you want an expression that will work for any r). I would definitely use cylindrical coordinates and just integrate in the z direction from 0 to some length l (the z length will cancel out later).

For the left hand side of Gauss's law you want to use a cylindrical Gaussian surface so that the electric field is always perpendicular to the surface. In this case [tex] \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = |E|A \hat{r}[/tex] where A is the area of the of the cylinder (again just use the variable l for the length in this area). After doing the integral to find [itex] Q_{enc}(r) [/itex] you should be able to equate the two sides and solve for E(r) for r<R. When r>R you need to think about what [itex] Q_{enc} [/itex] will be, but I think you have plenty here to start the problem.
 

What is an electric field?

An electric field is a physical quantity that describes the influence that a charged object has on other charged objects in its surroundings. It is represented by a vector, which has both magnitude and direction.

How is an electric field produced?

An electric field is produced by a charged object, such as a charged cylinder. The presence of a charged object creates a force field, which can exert a force on other charged objects in its vicinity.

What is a charged cylinder?

A charged cylinder is an object that has a net electric charge, either positive or negative, distributed along its surface. It can be thought of as a long, thin tube with a charge imbalance.

How does the electric field vary on a charged cylinder?

The electric field on a charged cylinder varies depending on the distance from the cylinder's surface. It is strongest near the ends of the cylinder and decreases as you move towards the center. The direction of the electric field also changes as you move around the cylinder's surface.

What is the relationship between the electric field and the charge on a cylinder?

The electric field on a charged cylinder is directly proportional to the charge on the cylinder. This means that as the charge increases, the electric field also increases, and vice versa. However, the electric field also depends on the size and shape of the cylinder.

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