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 do Gauss's law with NON-uniform charge density inside a non-conducting material

The vast majority of Gauss's law problems we do deal with uniform charge densities for non-conductors/insulators. These are the cases where there is charge inside the material, and therefore an electric field, and our job is to find the electric field inside. When the charge density, rho, is constant/uniform, the classic result is the field is linear with r. That is the result whether we have a sphere or a cylinder, and would be the same for gravity, electric fields, or magnetic fields (using Ampere's law).

But what about NON-uniform charge density, where rho depends on radius, r? What do we do with Gauss's law to find the electric fields inside these type of materials and objects? This video is an example of how to handle it. The gist is we need to set up an integral where we add the charges within skinny, hollow spheres of charge. Each little sphere has its own charge density value, and so the charge of each hollow shell is rho x dV. The trick is the dV = (4*pi*r^2)(dr), at least for a sphere. The dr is the small thickness of the hollow shell. The same idea holds for cylinders, where dV = (2*pi*L*r)(dr). I hope this helps!

How to Integrate and Find Electric Potential in Gaussian Problems

We have the task of integrating electric fields of gaussian systems in order to find electric potential, or simply voltages, of various regions in order to fully understand such systems. This is all leading to understanding how capacitors work, which is a primary component of all electronic devices. This is a multi-layer example, with a conducting ball surrounded by a couple thin shells. The reference point of V = 0 is 'far away.' I hope this helps...

How to Do Gauss's Law with NON-Conducting Materials

The previous video dealt with Gauss's law and conductors, where, when in electrostatic equilibrium, the electric field = 0 inside. Any net charge sits on the outer surface of a metal chunk. But, when we move to NON-conducting materials, things are more involved since there are NO de-localized electrons. It is OK to have net charge inside the material, spread throughout the volume, and therefore it is OK to have an electric field inside these materials. There would still just be static charge, and no currents. This is also the case where we would have charge densities. Check out how to do this for uniform charge densities, and I hope it helps while you review this topic.

How to Do Gauss's Law with Conducting Materials

Conducting materials separate themselves from non-conductors due to the metallic bonds. Those delocalized electrons that are free and drifting randomly throughout a metal are forced to move when an electric field is present, and currents are formed. However, with Gauss's law, remember we only can apply it to electrostatic situations. This is why charges, through polarization, rearrange and give NO E-field inside when we have static charge. Any net charge sits on the outer surface; this is the only way to have equilibrium. Check it out or review it.