How to do a NON-Gauss's law problem - A Partial Ring of Charge

Here is another example of a NON-Guass's law problem, a partial ring of charge. This is a combination of sticks and rings, the two classic cases where we might have to calculate something like an E-field or potential on some axis. In any case like this, we need to use the fundamentals: point charges, which we know how to do exactly. We break the problem into a bunch of little charges, find the small contribution, and then add them all up using an integral! Let's check it out, and I hope it helps.

How to do NON-Guass's Law Problem - A Stick WITH Ends

Gauss's law provides a relatively easy way to find E-fields for charged spheres, long sticks or cylinders, and large plates. The last two, of course, are approximations in the end, but ends, edges, and corners really make for a difficult math problem. When stuck with such a problem, we have no choice but to stick (heh, heh) with the fundamentals. That is, point charges. We know how to handle point charges, and physically that is all a charged object is - a bunch of extra charged particles. So we need to set up an integral and add up a bunch of small fields or potentials! Check it out.

How to find electric fields and potentials for systems of point charges

One of the most essential and basic systems we study includes systems of stationary point charges. These bring out the essence of what charges do, which is produce a vector quantity, electric field, and a scalar quantity, electric potential.

We need to remember that finding electric fields includes us forgetting about the sign of a charge when calculating the fields. Being a vector, let the picture tell you whether we are dealing with positive or negative directions of components. But for potentials, which are scalars, we DO need to include the signs of charges numerically since positive charges produce positive voltage, and negative charges produce negative voltage, and we just add up the values.