Place a single negative charge beside a wire carrying a current. While it stays still, it feels no force from the wire. The positive and negative charge in the wire are equal, so the wire as a whole is not charged.
Move the same charge in the same direction as the electrons in the wire, at the same speed , and it is pulled toward the wire. This force, felt only by moving charges, is the Lorentz force from the magnetic field.
To someone running with the charge, the charge is at rest. A magnetic field exerts no force on a charge at rest, so in this frame the pull toward the wire needs another explanation.
Inside the wire, fixed positive charges (the metal ions) sit at even spacings, and negative charges (free electrons) flow between them. The current is this flow of electrons.
Counted in the frame of the wire, the two rows have the same spacing. Writing the charge per unit length as and , they add to .
The spacing of the moving electrons is already a contracted value. In a frame where the electrons are at rest the spacing is times wider, and their charge per unit length is .
Move to the frame running at the same speed as the charge. Here the electrons are at rest and the positive ions stream backward at speed .
The ions, now moving, close up by length contraction, giving . The electrons, now at rest, spread out to .
The sum is , not . Seen from this frame, the wire is positively charged.
A negative charge sits at rest beside a positively charged wire. What pulls it toward the wire is the electrostatic attraction between positive and negative.
At distance from the wire the electric field is , so the charge feels a force of size .
In the frame of the wire, the current makes a magnetic field , and the moving charge feels . Using , the electrostatic force is exactly times this magnetic force.
A sideways force is measured times larger in the frame where the body is at rest. With this included, the forces in the two frames agree.
From the wire, a magnetic force; from the charge, an electrostatic force. The names differ, but both describe one event, the charge being pulled toward the wire, from two frames.
Electrons drift through a wire at about mm/s, so is only about . Magnetic forces can still be felt by hand because the charge in the wire is enormous, and even a tiny imbalance between positive and negative leaves a force behind.
Electric and magnetic fields are not separate things; they turn into each other when the frame changes. Maxwell's equations, with this turning included, keep their form under the Lorentz transformation.