Integrating the spherical shell
This page describes how to integrate a uniformly charges spherical shell in order to prove that it will look like a point charge from the outside but will have a zero electric field on the inside. The understanding of this is important because there are many objects that have a charge on the outside but have zero electric field on the inside.
The Main Idea
The electric field of a conducting sphere can also be found using Gauss' Law. Using Gauss's Law, you model a Gaussian surface of a sphere with radius r>R and the electric field will have the same magnitude, directed outward, at every point on the surface of the sphere.
This page serves more as a proof on understanding why there is an electric field outside the sphere, but not inside.
A Mathematical Model
Assume the spherical shell is centered at the origin.
Step 1: divide the sphere into (ring like) pieces. Imagine the sphere is sliced into several different rings. You must have these rings measure some distance 'theta' from the middle.
for example: [math]\displaystyle{ {ϴ, ϴ + Δϴ} }[/math] ... etc
Each ring contributes delta E at an obseration point some distance greater than r away from the center of the sphere.
Step 2: compute the distance of the ring from the observation location. (remember its :observation location - source) [math]\displaystyle{ {d = (0-Rcos(ϴ)} }[/math] find the amount of charge on each ring: [math]\displaystyle{ {ΔQ}={\frac{surface area of ring}{surface area of sphere}}=Q{\frac{2pi(Rsinϴ)(RΔϴ)}{4piR**2}} }[/math]
Note that the radius of the ring is Rsin(ϴ)' and its width is RΔϴ. The integration variable is: ϴ The Magnitude of ΔE is: [math]\displaystyle{ {ΔE}={\frac{1}{4piε0}}*{\frac{ΔQd}{d**2+((Rsinϴ)**2)**(3/2)}} }[/math]
={ΔE}={\frac{1}{4piε0}}*{\frac{r-Rcosϴ}{(r-(Rcosϴ)**2)+(Rsinϴ**2)**(3/2)}*{\frac{2pi(Rsinϴ)}{4piε0}}*{RΔϴ}</math>
A Computational Model
How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript
Examples
Be sure to show all steps in your solution and include diagrams whenever possible
Simple
Middling
Difficult
Connectedness
- How is this topic connected to something that you are interested in?
- How is it connected to your major?
- Is there an interesting industrial application?
History
Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
See also
Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?
Further reading
Books, Articles or other print media on this topic
External links
References
This section contains the the references you used while writing this page