How to do Biot-Savart for Loop of Current

A second example of Biot-Savart is with loops of current. We know that any current will produce a magnetic field, but a circular loop of current produces a linear magnetic field along its central axis. This is like the Helmholtz coil we use in the lab. Check out how to set this up and get an expression for that central axis.

How to do Biot-Savart for Straight Wires with Ends

We have done some work already with Ampere's law for finding B-fields created by long wires, solenoids (with no ends) and toroids. But in reality wires have ends, and we must use the real thing, Biot-Savart, to get those magnetic fields. Check this out for finding the integral for a straight wire with ends.

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 Magnetic Flux - Straight wire next to a loop

This is a classic type of problem, where a current in a straight wire is next to a rectangular loop, and we need to find the total magnetic flux through the loop. We need to break the loop into skinny strips of area, and find the flux through those skinny strips, then add them all up, i.e. integrate! Check it out to remind yourself.

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 Calculate Moments of Inertia with Integral - Sticks

When it comes to finding moments of inertia, the one thing we can find exactly is the inertia of a point mass, I = mr^2. Here, m is the mass, and r is the distance from the mass to the axis of rotation. But we run into some amount of difficulty when we have real objects that rotate and are made of countless point masses, i.e. atoms. How can we get the total moment of inertia?

We have an integral definition for I. What is is really telling us to do is break up the object into a bunch of small pieces of mass, dm. This is like saying break it up into a bunch of point masses, find each individual inertia, and then add (integrate) them all up to get the total. I will show how to do this with a stick in this video, which is the main case we would ever need to use the integral for class. Keep in mind that the other inertias we use for disks, balls, and so on, are found with this integral, too. I hope this helps. Keep in mind that I also have another video that shows how to use the parallel-axis theorem, which is a way around having to do the integration if you happen to know the moment of inertia of an object with the axis through the center of mass of the object.

How to Find Capacitance for Spherical and Cylindrical Capacitors

There are three shapes of capacitors in practice: parallel-plate, spherical and cylindrical. Conveniently, these are the three shapes we have for Gauss's law applications. We will use Gauss's law to find the E-field in the capacitors, then integrate the fields to get the potential difference across the capacitor, and then use our definition of capacitance, C = Q / V, to get the capacitance expressions. Let's take a look at two of the three, spherical and cylindrical capacitors.