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As far as I understand, electric field lines are only a way to "visualise" electric field. There is no specific rule for the number of field lines to be drawn for a particular field. If it is so, then how does electric flux represent the "number of field lines passing through a surface"? I have read about the analogy of measuring the "number of particles" that flow through a particular area in a fluid, but I am unable to make any connections between the two.

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A mechanical force can be drawn with a line segment, to represent its module with length, but in electrical field, forces are not defined, lines are only "potential interaction forces", in the sense that, if you put a charge there a force will arise from the interaction, so fields lines have no definite ends (except sources-sink in charges) so how to represent intensity when the line length isn't available? Using number of lines is the preferred way. Why?

Because of electric flux, that is an important physical quantity, is the field passing through a given area in perpendicular direction, so representing intensity with lines is useful to see flux, counting lines per area let you visually calculate flux and compare the effect of different fields and areas.

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As you mentioned, the number of field lines depend on how much we draw and does not seem to relate with flux. And actually, flux has nothing to do with number of field lines passing through the area. We can say, as for a physical interpretation of flux, that it is propotional to the number of field lines passing through the area (normally) .

Flux is a different quantity. It is just like Area. we cannot physically interpret what area is. We may say that it is the number of parallel lines with unit separation that can be drawn in a region. But we may reduce the unit separation and include more lines... we may only say it is proportional to the number of lines drawn.

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If you divide space into many discrete little cubes and draw the average $\vec{E}$ inside each cube in the center. You get a vector plot and it is useful. If someone give you only this vector plot and tell you the units you can use it to calculate the flux approximately.

The electric field lines is a different picture than the vector plot, I think it's closer to a streamline plot. You can systematically construct a streamline plot given the expression of $\vec{E}$. In this sense, I think you can also calculate approximate flux from the streamline plot.

Although the electric field lines are often used with illustration purposes, I think one can actually systematically construct them from the function of the field and make them practically useful for calculating flux. (I haven't tried and so don't know if it would be hard)

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