# Direction of gravitational field given equipotential lines

I've attached the question as an image below as it's a graphical question.

It simply states:

"The diagram shows equipotential lines near a group of asteroids. Which arrow shows the direction of the gravitational field at X? (North, South, East or West)"

From my own interpretation, I would've guessed that as the direction of the gravitational field is usually in the direction from a lower (magnitude) potential to a higher potential, i.e. a mass placed at X would be attracted towards the $$-3.5$$x$$10^{-9}Jkg^{-1}$$ equipotential line from the $$-1.5$$x$$10^{-9}Jkg^{-1}$$ equipotential line, so that the direction would be pointing north.

However, the answer is C (west). I don't understand this.

Any help would be much appreciated, thank you!

I assume you dont know much about differentials, derivations and such, so i will be little sloppy.

The point X has no idea where the line $$-3.5*10^{-9}Jkg^{-1}$$ is, if i may say it this way. The point X knows only about its immediate surroundings.

You are right, that mass placed at X will be attracted to the most negative potential. But only to the most negative potential next to the point, not everywhere in the universe.

To be more precise: At the point X, you know the value of the field + you know how the field is changing at the point. When you move North in very small (infinitesimal) amount, you will basicly still be at the equipotential line, so the potential basicly doesnt change at all. However if you move West, then you are going as much away from the equipotential line as you can, and therefore the potential will change the most. Thus, the mass will move West, because that is the direction in which field changes the most quickly to negative values (for very very small - infinitesimal - displacements, because you dont know what is far away from the point X)

• Ahhh, this makes a lot of sense! It's much better to think of it exactly at X, moving very very small amounts, because I presume the gravitational field direction changes depending on where a mass is in the field (in this case)... always wanting to move to where the most negative potential is at that point in time. Thank you very much! :) – MKY6 May 13 at 15:37

Things tend to fall downhill i.e to the position of lower potential: $${\bf F}= -\nabla \phi.$$

• But as it's negative potential, won't a mass placed at X be attracted to the more negative potential? i.e. the gravitational field will do work on the mass to move it from the less negative to the more negative potential? The mass would otherwise need to do work against the field if it wanted to move to a more positive potential, so doesn't that mean the direction of the field is from less negative to more negative? – MKY6 May 13 at 15:20
• I've just noticed something... if what I'm saying is right, at X it will be going west! But then if it were to go from the -3.0 to the -3.5 potential would the field there be pointing north? So does that mean, in this field of asteroids, the gravitational field direction is changing? – MKY6 May 13 at 15:21
• Exactly! The potential gets more negative as you move to the left: from -1.5 to -2. – mike stone May 13 at 15:22
• Great! Thank you! Just to clarify... if, say, they placed a point Y on the -3.0 equipotential line and asked the same question, would the answer be pointing north? – MKY6 May 13 at 15:26
• Yes. Roughly..... . – mike stone May 13 at 15:32