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I heard that a magnetic field only creates a force on a moving charge, not a stationary one. However, what if the solenoid containing the magnetic field was moving through a stationary charge. Would that create a similar force? Thanks!

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Yes. A changing magnetic field generates a changing electric field, that acts in the charge ($\mathbf F = q\mathbf E$).

Another way to see the same effect is moving to the frame of the solenoid. Now the charge is moving and a magnetic force acts in the charge ($\mathbf F = q\mathbf v \times \mathbf B$)

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    $\begingroup$ I would add that if the charge is on the axis of an idealized solenoid, there would still be no force on the charge because the E-field would be 0 there. $\endgroup$ – Puk Jun 14 at 22:33

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