Field strength at point using field lines

Question

I have recently read that the electric field strength can be compared between 2 points from the relative field line density but I am having trouble in applying the concept. What I find confusing is the field line density.

I feel the main problem is that (according to me) we need to compare how close the field lines are in an area around the point. But what is that area going to be? Is it going to be a circle centered at the point? If yes, will it be a small circle or a circle of any radius? It would be of great help if somebody could explain the concept and relate it to the diagram I have drawn:

Thanks

Answer 1

If yes, will it be a small circle or a circle of any radius?

If you want to make a quantitative estimate from a line diagram that truly works, then this needs to be understood as a limit, taking a sequence of circles with ever-decreasing radii. And, to make sure that you always have field lines crossing your circle, at each step you need to increase the total number of field lines.

If you don't have access to that limiting procedure, then you'll only be able to get a rough estimate (and probably quite a coarse one) on the relative field strength, by taking as large a circle as you can while still having a reasonably homogeneous field strength (i.e. field lines equidistant and parallel) within both circles. The degree to which you're able to achieve those properties within both circles will then dictate how good of an approximation you're able to make.

For more details, see my answer to Why does the density of electric field lines make sense, if there is a field line through every point?

Answer 2

I have recently read that the electric field strength can be compared between 2 points from the relative field line density but I am having trouble in applying the concept. What I find confusing is the field line density.

I sympathize with you. To me, there is no accurate definition of that density. At best it can be used as an illustrative device, at a (rough) intuitive level.

Moreover, consider that drawings are 2D, whereas a physical vector field is 3D. It's not just a drawing difficulty, it's a conceptual one. Think of the simplest case, a charged sphere. How would you arrange a finite number of field lines starting from that surface? If you meant to compute a limit increasing that number, what rule would you adopt?