Charged Cylinder

Electric Field

Topic reserved by Jennifer Burkhardt

The electric field of a uniformly charged cylinder of length [math]\displaystyle{ L }[/math] and radius [math]\displaystyle{ R }[/math], with [math]\displaystyle{ L≫R }[/math] can be found through viewing the cylinder as a collection of long uniformly charged rods forming a circle or a series of uniformly charged rings stacked on top of another. Before determining which method to use, the point of observation must be decided.

Uniformly Charged Rods=

If measuring the electric field of a cylinder from an observation point that is at the center of the cylinder, the method of rods would be most useful.

Mathematical Model

The location vector from the center of a rod can be determined in terms of the angle θ.

[math]\displaystyle{ \vec{r} = \lt 0,0,0\gt - \lt 0,R*sinθ,R*cosθ\gt = \lt 0,-R*sinθ,-R*cosθ\gt }[/math] [math]\displaystyle{ \hat{r} = \frac{\vec{r}}{r} = \lt 0,-sinθ,-cosθ\gt }[/math]

The amount of charge on one of the rods, assuming that the cylinder is complete, is given by:

[math]\displaystyle{ dQ = Q\frac{dθ}{θ(total)} = \frac{Qdθ}{θ(total)} }[/math]

[math]\displaystyle{ dθ }[/math] is the angular width of one rod, and [math]\displaystyle{ θ(total) }[/math] is the angular extent of the cylinder.

The contribution of this individual rod to the total electric field,, assuming that the cylinder is complete, is:

[math]\displaystyle{ \Delta \vec{E} = |\Delta \vec{E}|\vec{r} ≈ \frac{1}{4π\epsilon_0} \frac{2dQ}{L} \frac{1}{R} \lt 0,-sinθ,-cosθ\gt = \frac{1}{4π\epsilon_0} \frac{2}{LR} \frac{Qdθ}{θ(total)} \lt 0,-sinθ,-cosθ\gt }[/math]

To find the total electric field from all of the rods, the integral of the electric field will have to be taken from the x, y, and z components separately.

[math]\displaystyle{ E_x = \int\limits_0^{θ(total)}\ \frac{1}{4π\epsilon_0} \frac{2}{LR} \frac{Qdθ}{θ(total)} * 0dθ = 0 }[/math]

[math]\displaystyle{ E_y = \int\limits_{0}^{θ(total)}\ \frac{1}{4π\epsilon_0} \frac{2}{LR} \frac{Qdθ}{θ(total)} * (-sinθ)dθ = \frac{1}{4π\epsilon_0} \frac{Q}{LRπ} cosθ|_{0}^{θ(total)} }[/math]

[math]\displaystyle{ E_z = \int\limits_{0}^{θ(total)}\ \frac{1}{4π\epsilon_0} \frac{2}{LR} \frac{Qdθ}{θ(total)} * (-cosθ)dθ = \frac{-1}{4π\epsilon_0} \frac{Q}{LRπ} sinθ|_{0}^{θ(total)} }[/math]

Uniformly Charged Rings

If measuring the electric field of a cylinder from an observation point that is away from the cylinder in the z-direction (assuming that the circles the cylinder is composed of reside in the x-y plane), the method of rings will be most useful.

Computational Model

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Examples

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Simple

Find the net electric field at the center of a half cylinder, projected on the z-axis, that is 2 meters long, has a radius of .05 meters (2≫.05), and has a 3 Coulomb charge.

Solution

[math]\displaystyle{ dQ = Q\frac{dθ}{π} = \frac{Qdθ}π} }[/math]

[math]\displaystyle{ \vec{E}_{net} = \lt x,y,z\gt }[/math]

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