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If both electric and magnetic fields attenuate with distance by the inverse square law, what difference does it make that the latter's field lines are 'closed'?

And if a magnetically-induced electric field has 'closed' field lines, rather than open, so what? How can you tell?

You never learn in school precisely what those field lines mean in the 'real' world.....

Or maybe I'm just an idiot...

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The statement that magnetic field lines are closed essentially says that there is no magnetic monopole where the fields lines can start or end on. And in fact, because of this, bounded current and charge distribution cannot produce a magnetic field that goes like $r^{-2}$ at large distance. For your second question, a closed electric field line will mean that a charge can go around the loop and gain energy, i.e., there is an EMF, which can be easily measured.

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Magnetic field lines are generally not closed. That they should be closed is a very common misconception in the teaching of physics. See e.g. here https://physics.aps.org/story/v24/st24

From Maxwells equations we have that div B=0, this is a local statement that field lines dont have a beginning nor an end, this does not imply the global statement that they close.

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Magnetic field lines are closed because there are no magnetic “charges” (usually called magnetic monopoles) analogous to electrical charges like protons and electrons. When electric field lines end, they always end on an electric charge.

The apparent absence of magnetic charges makes for an unattractive asymmetry in Maxwell’s equations. Some physicists think that magnetic charges can exist but are now extremely rare. They may have been common in the very early universe, before cosmic inflation occurred.

Electromagnetic field lines help you visualize the direction and magnitude of the forces that the field exerts on charges. For an electric field, each field line points in the direction of the force it exerts on a positive charge. Where the field lines are closer together, the force is stronger, and where they are farther apart the force is weaker.

For a magnetic field, it’s more complicated, because only a moving charge feels a magnetic force. The magnetic force is perpendicular to both the magnetic field line and the charge’s velocity. Again, field lines that are closer together indicate a stronger force.

I prefer to visualize electric and magnetic fields as two little vectors at each point in space. You can get the field lines just by “connecting the arrows” from point to point.

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