Drawing a gravitational field inside a planet

[MalachiK]

For a point mass acting as a source of a g field we draw lines that point inwards from infinity. Where we have an extended body like a planet, I've always drawn the field lines terminating on the surface. However, I've seen a question in an exam paper that implies that it's okay to continue drawing the field inside the planet with the lines getting closer together before meeting in the middle.

Surely this is what the field looks like for a singularity. Inside the planet, the field falls away linearly to zero at the centre. This being the case, it seems to me that you can't just keep drawing the field lines getting closer together as increasing the density of the lines represents an increasing field strength.

What bothers me is that the field lines inside the planet all still need to point towards the centre, so it's difficult to imagine the patten. My best guess is that because the lines need to end on a mass, you get a picture where the lines are always radial and get closer together inside the planet BUT, as we move inwards, some of the lines terminate and the overall density of filed lines goes down.

Is this right?

 

[pbuk]

Is this right?

Nearly - but you have contradicted yourself:

the lines ... get closer together inside the planet

... the overall density of filed (sic) lines goes down.

Do you know which is correct (you will need to make an assumption about the distribution of mass within the planet)?

 

[MalachiK]

I don't think there's a contradiction. I'm saying the direction is towards the centre but some lines terminate before you get there.

 

[A.T.]

the density of the lines represents an increasing field strength.

Note that for a 2D picture this already breaks down on the outside, where the field strength falls off with 1/r², not linearly as the distance between radial lines.

 

[MalachiK]

I don't think there's a contradiction. I'm saying the direction is towards the centre but some lines terminate before you get there.

Or rather, it would be better to say that this is the contradiction that I'm asking about. How can the lines continue to point inwards and not get closer together UNLESS some of the lines terminate on shells inside the planet.

This is the idea for which I am seeking confirmation.

 

[ZapperZ]

For a point mass acting as a source of a g field we draw lines that point inwards from infinity. Where we have an extended body like a planet, I've always drawn the field lines terminating on the surface. However, I've seen a question in an exam paper that implies that it's okay to continue drawing the field inside the planet with the lines getting closer together before meeting in the middle.

Surely this is what the field looks like for a singularity. Inside the planet, the field falls away linearly to zero at the centre. This being the case, it seems to me that you can't just keep drawing the field lines getting closer together as increasing the density of the lines represents an increasing field strength.

What bothers me is that the field lines inside the planet all still need to point towards the centre, so it's difficult to imagine the patten. My best guess is that because the lines need to end on a mass, you get a picture where the lines are always radial and get closer together inside the planet BUT, as we move inwards, some of the lines terminate and the overall density of filed lines goes down.

Is this right?

Have you ever done the Gauss's law equivalent for gravitational field?

Zz.

 

[MalachiK]

Nope, only ever used it to derive the radial electric field. But since you mention it, I'll go and look it up.

Thanks

 

[ZapperZ]

Nope, only ever used it to derive the radial electric field. But since you mention it, I'll go and look it up.

Thanks

Excellent. Please follow up if you get stuck. The math and the physics are similar to solving Gauss's law for a spherical charge.

Zz.

 

[pbuk]

I don't think there's a contradiction. I'm saying the direction is towards the centre but some lines terminate before you get there.

I see what you mean; my point was that the lines need to terminate rapidly enough to decrease their density because (for a uniformly dense planet) gravity decreases with depth.

Coming back to the original point, what level was the exam? If it was at a level where understanding of gravity within a massive body is not required then lines converging to a point within the planet, although strictly incorrect, would not lose any marks.

 

[sophiecentaur]

I don't think there's a contradiction. I'm saying the direction is towards the centre but some lines terminate before you get there.

This is the problem when you take an abstract concept like the field and try to model it in a very concrete way with 'field lines'. The model can easily let you down, as it does here. There are no lines, so they don't have to follow any rule that your intuition might suggest. You have to be prepared to chuck all models away at some stage and not to flog a dead horse.

 

[MalachiK]

This is the problem when you take an abstract concept like the field and try to model it in a very concrete way with 'field lines'. The model can easily let you down, as it does here. There are no lines, so they don't have to follow any rule that your intuition might suggest. You have to be prepared to chuck all models away at some stage and not to flog a dead horse.

I agree. What started me thinking about this is that the OCR A-Level Newtonian World paper for Jan 2010 has a mark scheme that (vaguely) implies that it's valid to stick radial field lines all the way into the centre of a planet. I was using this as a mock exam last week and had a 'cognitive dissonance' moment trying to reconcile my understanding of how physics hangs together with this suggestion. Since it was in an A-level paper I thought there might be something I was missing and that this could be understood at the level of pre-university study.

Have you ever done the Gauss's law equivalent for gravitational field?

I've looked over Gauss's Law for gravity and it seems exactly like how you recover the Coulomb force for a point charge. I thought about this for a while. If the flux is proportional to the enclosed mass and the enclosed mass falls off as we move inside the planet then it simply must be the case that taking smaller and smaller shells gets us less field though the surface each time. This is exactly what you'd expect from the Newtonian way of thinking, but the idea of decreasing gravitational flux is more germane to my question. I am content.