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In explanations of electrostatic induction of a conductor $B$ by the charge on another conductor $A$, there is often reference to the field lines.

For example, in a capacitor the electrostatic induction is said to be "complete" because a charge $q$ on plate $A$ makes a charge $-q$ appear on the opposite plate $B$.

This is explained saying that "it happens because all the field lines starting from the conductor $A$ with charge $q$ end up on the conductor $B$, where induced charge appear".

Besides making me visualize the phenomenon, I do not understand the physical role of field lines in electrostatic induction: for example, how is the number of field lines starting from $A$ reaching the conductor $B$ related to the amount of charge induced?
Are the fields lines not reaching $B$ completely unrelated to the phenomenon of induction on $B$, or do they play any role?

And, in general, how to interpet/understand the role of field lines here?

(In picture the field lines actually start from point charge instead of a conductor, but the question is the same). enter image description here

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Field lines do not actually exist. They are a visual representation of the field. They do not have any "physical role" in electrostatic induction or any other phenomenon.

Perhaps what you are asking is "What are they representing?"

The direction of the lines represents the direction of the electrostatic force on a small +ve test charge placed at that point. The density of field lines relates to the strength of the electric field. Field lines cannot intersect or overlap. They can only start on a +ve charge and end on a -ve charge.

Where they start or end on a conductor, there may not be an actual electron or missing electron at that point, only an excess of charge on average. They are always normal to any conductor where they meet the surface, and start or end there. A higher density of field lines at the surface of a conductor indicates a higher density of surface charge.

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