Here's an attempt at a qualitative explanation.
Suppose we have a very long, straight wire with current. Imagine the current is due to motion of both positive and negative charge carriers.
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The positive charges are shown in red and move upward. The negative charges are blue and move downward at the same speed. The charge density is the same for the positive and negative charge. The net current is upward. The magnetic field in the region to the right of the wire points into the page. So, a positive point charge that moves away from the wire toward the right will experience a magnetic force upward:
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If we go to the rest frame of the moving charge ##q##, the upward force on ##q## must be due to an
electric field. We want to see how this happens. In the rest frame of ##q##, the wire moves toward the left. So, the positive charges in the wire move upward and to the left. The negative charges in the wire move downward and to left. This is shown in the figure below. In the small region of the wire marked ##A##, the red arrow indicates the velocity of the positive charge carriers, the blue arrow denotes the velocity of the negative carriers.
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A uniformly moving point charge has a radial field that is stronger in directions perpendicular to the velocity direction. This relativistic effect is very small for non-relativistic speeds, but it's important here (I think).
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The pattern of the field lines for the carriers in region ##A## are shown schematically below:
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Finally, consider the diagram below which shows symmetrically placed regions ##A## and ##B## of the wire. The electric fields produced by the carriers in ##A## and ##B## at the location of the charge ##q## are shown.
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The red arrow ##E_A## denotes the field at ##q## produced by a positive carrier in region ##A##. The blue arrow ##E_A## is the field produced by a negative carrier in region ##A##. The blue field ##E_A## is stronger than the red field ##E_A## because the direction from ##A## to ##q## is closer to being perpendicular to the direction of the velocity of the negative charge carrier in ##A## than to that of the positive charge carrier. Similarly, the red ##E_B## and blue ##E_B## are the fields at ##q## due to the positive and negative carriers in ##B##, respectively. Here, red ##E_B## is strong than blue ##E_B##.
When you add the four field vectors at ##q## you get a net electric field that is upward, parallel to the current.