Gravity and Gauss's law: Inside the Earth

Gauss's law for gravity can be used primarily to show what happens if the earth or other really large objects had different shapes.  We would find that spheres and point masses have 1/r^2 behavior, cylinders have 1/r behavior, and flat objects would have a uniform, constant force of gravity.

But Gauss's law can be used to figure out what happens inside objects as well.  There are two really fun cases to consider: i) a hollow earth, and ii) a solid earth with uniform density.  These situations are normally considered for solid charged objects (nonconductors) in electricity and magnetism units, but the same math and concepts can be used for gravity, too.

The key to Gauss's law is that the strength of the gravitational field at some radius from the center of the earth depends on how much mass is inside that radius.  If the earth is hollow, this means that once you drill a hole into the earth and jump in, because there is no mass inside then there would be no gravity inside!  You would be truly weightless and would simply float around inside a hollow earth!!

Because the earth is not hollow, we will not have a chance to test that prediction from Gauss's law.  But a solid earth can be analyzed.  If we assume the earth has a uniform density, and we were to drill a tunnel all the way through the center of the earth to the other side, what would happen if you jumped into that tunnel?

Check out the video to see more details, but the answer is you would oscillate back and forth through the earth as if you were attached to a spring!  The motion would turn out to be the same as that of something in simple harmonic motion!

How to set up Integrals for Gravity for Rings and Sticks

We are used to doing gravity problems with Newton's law of universal gravitation, F = -GMm/r^2, for things like planets and stars and point masses.  But we also can show that the shapes of objects matters for how gravity behaves, such as the difference between spheres (1/r^2), long cylinders (1/r) and large flat objects (uniform gravity).  This comes from Gauss's law for gravity.

However, once the shapes become even more different, such as rings and sticks with actual ends, Gauss's law does not work and we need to figure out how gravity behaves then.  This is when we need to break an object into point masses, use Newton's law of gravity for each small mass, and then add up all the results to get the total...this is what integrals do for us!  Check this out, with the focus being how to set up the integrals.  We are not so focused on the final results, but rather the thought process and setup.

Why is Gravity 1/r^2? Gauss's law and Gravity

Gravity is obviously a big deal in physics and in life.  But in class we are used to just one idea, which is Newton's law of universal gravity, F = -GMm/r^2.  This is usually just handed to students and we start using it to solve all sorts of problems, but here I wanted to justify WHY gravity behaves this way, with the inverse-square law of 1/r^2.  It turns out it is a result of geometry!

This video attempts to introduce the notion of Gauss's law, which is normally a big topic in electricity.  But it can be applied to gravity, as well.  Gauss's law is a formal way of defining something called flux, or the amount of stuff flowing through an area.  Mathematically, flux = (stuff)(area).  In our case, the 'stuff' is the gravitational field that all particles and massive objects creates all around itself, and will create the forces we experience when two or more objects are near each other.

We look at spheres, cylinders, and flat objects.  Gauss's law then shows us that the gravitational fields will behave differently with distance from the object.  Gravity is NOT always 1/r^2!!  This is just for spherical objects and point masses.  If the earth were cylindrical, for instance, gravity would fall off more slowly, as 1/r.  And if earth was flat, then gravity would be uniform, or constant, no matter how close or far away you were!  Check out what Gauss's law looks like and where these results come from.

Note there are how to videos for Gauss's law in electricity, specifically for electric fields, on our class blog.

How to Think About Band Theory

In chemistry, we learn about the three main types of materials in terms of their electrical properties: conductors (typically metals like copper or gold), non-conductors (most materials with covalent bonds), and semiconductors (something like silicon).  But how and why are there different behaviors - all atoms are made of the same pieces, so why the variety of materials?

The answer can begin to be found with a relatively simple model called band theory.  This uses the concept of fixed energy levels that you first learn about studying the Bohr model of an atom in chemistry.  Bound electrons of atoms can only have specific, fixed energies.  But this picture gets muddied when you start thinking about real materials with lots of atoms - all these atoms interact with each other because they are all made from electric charges.  These interactions cause the energy levels of individual atoms to shift slightly, where some increase in energy and others decrease.  Collectively, the energy levels blur and form more of a continuous energy band.  Specifically, two bands form, one where the bound electrons exist (valence band) and one where they have enough energy to move freely through the material (conduction band).  To be a conductor requires many free electrons in the conduction band.

Non-conductors have a large gap between the valence and conduction bands; semiconductors have small gaps between the two bands, and conductors have an overlapping between the two bands, meaning all sorts of electrons are free to move through the material.  Some call this the 'sea of electrons' in a conducting material.  Check out the model!  Also, feel free to play with a PhET simulation where you can have 1 - 10 atoms (or wells) near each other, and see how the energy levels change to begin forming bands.

A Good Physics Blog

Find all sorts of information about a large variety of physics topics at the site http://www.physicscentral.com/buzz/index.cfm.  This is run through the American Physical Society, and has links to specific areas of physics as well as new applications of physics in everyday devices, and much more. Check it out!

The Power of Black Holes and their Jets

Black holes are among the most powerful and weird objects in the universe.  One aspect of these critters is that they eject jets of radiation and particles out from the accretion disk.  New studies suggest these jets come out at some 99.9% the speed of light and can extend for millions of light-years!  This constitutes an amazing amount of energy.  Check out an article here.

How to do Multi-loop Circuits

There is a type of resistor-battery circuit that is strange to the degree we cannot find the total resistance due to the presence of multiple batteries.  The trick, then, is to treat each part or loop of the circuit as an independent circuit, and use Kirchhoff's voltage rule to set up loop equations: the battery voltage = sum of voltages of components in the circuit. Check it out!