Physicopoeia

φυσικοποιΐα
THINGS I WISH I'D KNOWN WHEN I STARTED TEACHING PHYSICS

Gravitational Field Lines... inside a planet?

I have a few bugbears about how magnetic field lines are often drawn poorly (see this, for example), and there can be problems with gravitational field lines too. If given a picture of a planet and asked to draw the gravitational field lines on it, you would probably draw radial lines directed inwards to the centre of the Earth. That's fine, but what do you do when the field lines meet the surface of the Earth? There's no perfect solution to this, because gravitational field lines don't apply well as a pictorial tool to this situation. Most sources simply sidestep the issue and make them stop on the surface of the Earth (here's an image search which shows how common this is). Unfortunately, this implies that there is no gravitational field below the Earth's surface, which any miner can tell you isn't true.

 

So why not continue the field lines radially inwards until they meet at the centre of the Earth? Well, since they would continue getting closer together, this would imply the gravitational field gets ever stronger as you approach the centre of the Earth whereas, in fact, it gets weaker. So, you're wrong if you draw them and wrong if you don't! I've seen one textbook try to get around this by continuing the field lines inside the Earth but making them get steadily fainter towards the centre, which has some merit to it but isn't really something you can reproduce easily on a board or piece of paper.

So what do I do? I start with the usual picture of the field lines stopping on the surface and then ask the class what happens inside the Earth. This leads nicely to a discussion of the problem and the conclusion that field lines are a very useful tool in many ways, but have their limitations.

WHAT ABOUT ELECTRIC FIELD LINES?

The same issue doesn't apply to electric field lines, because inside a hollow charged conductor, for example, there is no change in the electric field so it would be quite correct to draw the field lines starting or finishing on the surface of a hollow charged object.

HOW DOES g CHANGE INSIDE THE EARTH?

Many exam boards want pupils to sketch or plot how g changes as you move away from a planet, simply expecting them to show they understand that it decreases as an inverse square law.

An interesting discussion (and calculations) can result from asking what happens if you were to travel inside the Earth. It should be fairly clear that at the exact centre of the Earth should be zero, since any mass there would be equally attracted in all directions - assuming the earth is spherically symmetrical which, to a good approximation, it is. But how does it vary between the centre and the surface?

There are at least three different answers to the question, depending on how accurate an answer you want:

  •  The simplest solution is based on the assumption that the Earth's density doesn't change as you go deeper - this predicts that decreases linearly from 9.81 Nkg⁻¹ at the surface to zero at the centre.
  • Given the huge increase in pressure as you go deeper inside the Earth, it's not surprising that the density is not, in fact, the same at all depths. If you assume that its density increases linearly with depth, you get a curved decrease.
  • Finally, if you account for the density variations with the different layers inside the Earth (mantle, core and so on), you get a rather complex pattern, though it's fairly similar to the linear-density-increase model.

Here's a graph from Wikipedia, comparing all three models.

You may well have come across the classic 'what happens if you drop an object through an imaginary tunnel through the Earth' situation, and showed that it undergoes SHM. In fact, it would only undergo SHM if the Earth were of uniform density. Since it isn't, the situation is, unfortunately, even less realistic than it seems to start with. It's still worth showing to pupils though, if only because it's such a fascinating solution which draws together several bits of Physics that we usually teach separately. If you don't already, take it one stage further and show that the SHM would have a time period of just under 90 minutes which is more or less what satellites in low Earth orbit have - this is a wonderful way of reminding pupils that circular motion and SHM are very closely related.