Find the electric field of a uniformly charged ring of radius 'r.'
I think it would require spherical integration and some trig (i.e. it's probably going to be long and messy).
First, I would draw a picture. Mine has the ring on the x-y plane centered at the origin.
Next, I would pick a random point on the graph. I drew one off to the side in the upper half of the space. I would then draw a vector from the origin to the point. I call this position vector a. Next, I will draw a position vector from the origin to a point on the ring and call it b. There is a third vector you can draw now, one from the end of b on the ring to the random point (the end of a) Let's call this resultant vector c. Now you should have a triangle.
To find the electric field, you need to first find the potential at that point. To do this, you need to find the contribution from the entire ring. This means you need to integrate along the ring.
The potential due to one small infintesimal portion of the ring is a function of the length of c. To find the length of c, you'll need to do some trig (law of cosines). Since you picked a random point, you'll need to generalize the coordinates. The length of c will undoubtedly be affected by what part of the ring you chose. Therefore, you must integrate your expression around the ring. Then, take the negative gradient to get the electric field.
I can't really go into much detail without actually solving it. Perhaps there's an easier way that someone knows.
I assume you mean to find the electric field due to a uniformly charged ring of radius R.
On the axis of the ring, that is a standard elementary textbook problem.
It is easier to first find the potential, and then the E field will be the gradient of the potential.
To find the potential off the axis, you first expand it in a power series.
Then the power series can be related to a Legendre polynomial expansion in cos\theta to find the potential off the axis. (I assume this is what you meant by "over all space".) This off-axis problem is done in some advanced texts.
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