# Understanding field representation of force [duplicate]

I am reading the book The Evolution of Physics. I have a doubt in the topic "The field as representation". In this topic authors give the example of gravitational force represented as a field. In the following image the small circle represents an attracting body(say sun) and the lines are the well known lines of force of the gravitational field.

It is said that the density of the lines of force in space shows how the force varies with the distance. Let us consider a finite volume $\Delta V$ in the vicinity of sun. Now the number of lines of force passing through this is finite but there are infinite points in this $\Delta V$ volume.
$1.$Is there any gravitational force acting on those points through which no line of force passes.
$2.$ If the gravitational force acts on all the points contained in $\Delta V$ shouldn't there be infinite lines of forces passing through $\Delta V$.

$3.$ If it is supposed that there are really infinite lines of force passing thru $\Delta V$ then how to decide the density of no of lines won't it be infinite.

Please cite some canonical references which explains the 3 different points I've mentioned in your answer.

Thank you.

• Field lines are just an aid, equations are the important things. What does $V=-GM/r^2$ tell you? – jinawee Jan 15 '14 at 13:09
• @jinawee The equations $g=GM/r^{2}$ tells me g is continues so $mg$ is also continues. This implies that force exists at all $\infty$ points. But i found field representation a bit ambigous thats why i asked this question. And i think field lines are as important as the equation ,infact the book strongly endors this fact. – user31782 Jan 15 '14 at 13:20
• – Qmechanic Jan 15 '14 at 13:29
• It is important to note that "fields lines" are a visualization of the field. The field is the real thing and the lines are only a map of the field. See for instance physics.stackexchange.com/q/80912. – dmckee --- ex-moderator kitten Jan 15 '14 at 19:39
• @dmckee you are using a different terminology. In the book i am reading the word "field" is used interchangeably and synonymously with "lines of force". so "fields lines"= "field". – user31782 Jan 16 '14 at 8:50

There are field lines going through every point. Obviously, it's only possible to draw finitely many of them. The picture contains an implied "and so forth" --- that is, you are supposed to assume that the infinitely many un-drawn lines look qualitatively like the drawn ones.

Analogy: Think about a topographic map, which shows lines (or curves) of elevation. There's a curve labeled "elevation 1000 feet", one labeled "elevation 2000 feet", one labeled "elevation 3000 feet", etc. Many points are on no curve at all. You're not meant to conclude that those points have no elevation; you're meant to conclude that they have elevations which can be (more or less) inferred from the curves that are drawn.

• Would you cite some reference about your answer, i want to make it sure that we are supposed to have infinite lines of force. – user31782 Jan 15 '14 at 14:04
• @anupam This is right. Just think what would happen if your supposition were correct: to scape gravity you could step aside field lines. It's like a phase portrait of a differential equation there are infinite uncountable solutions, but you just plot some of them. – jinawee Jan 15 '14 at 15:32

No field representation of force, the representation you are seeing I.e. the lines of force representation indicates relative field/force strength with respect to distance, as you can see at different distances from the source the field density(no of lines/ area) is different.

The lines of force representation does not represent where the force reaches or not, it simply designates relative strength and the direction of force on a conventionaly defined unit particle such as point charge in electrostatics used to determine field is conventionally taken to be positive and field lines are drawn in accordance to the direcction of force it feels.

For further reading you can refer to FIELD LINES, it has everything from the definitions to their physicsl significance, and was just a google search of "field line" away !

• i am commenting here 'cause i cant on your other answer. You said that "It does not matter if you make n infinite..." What did you mean? Are you saying $n=\infty$? Please explain a bit more on this point. – user31782 Feb 2 '14 at 3:39
• Yes I meant that, what's there to explain about it ? Please tell what is your query and only then I can solve it. Also why cant you comment there ? – Rijul Gupta Feb 2 '14 at 8:54
• By definition $\dfrac{\infty}{\infty}$ is undefined. In mathamatics there are exactly infinite points in a finite space. All the people are saying $n/n=1$for $n=\infty$ which is wrong. This is what my question is all about. I don't have the privilege to comment everywhere. – user31782 Feb 2 '14 at 14:00
• Infinity is a pretty ambiguous mathematical construct taking an $\infty/\infty$ approach can be also seen as $0/0$ ofcourse this makes no sense as if you draw no lines then field line density has no meaning ! Try to see $\infty$ as a really big real number (bigger than all numbers) and not as a mathetatical construct meaning $1/0$ then its physical significance would make sense. – Rijul Gupta Feb 2 '14 at 17:50
• "Infinity is a pretty ambiguous mathematical construct". Rijul mathematics is never ambiguous. Infinity and zero are well defined. There is a great difference b/w $n\to \infty$ and $n= \infty$. Physics accepts the mathematical definition of a continuum. So there are infinite points in a finite space. one infinty is not greater than the other. Can you say there are more points between $1 and 2$ as compared to that of between $1 and 3$ on a straight line with $1,2 and 3$ bieng the respective coordinates of three different points. In language of maths $1,2 and 3$ represents cuts [cont.] – user31782 Feb 3 '14 at 11:42