File:BfromE.gif

English: Electostatic-field (right) causing the magnetic-force (left) of a neutral-wire's current on a moving-charge: This figure shows two views of the frame-invariant proper-force (green-arrow) on a positive moving-charge (blue-dot), due to a current of negative electrons (red-dots) in a neutral wire (shown in the left panel). The electrons in the wire are moving in the same direction as the moving-charge, and (for simplicity) at the same speed.

Notice that the magnetic force on the moving-charge in the frame of the neutral-wire (left) results from an electric field in the frame of the moving-charge itself (right). This is true in general.

In standard symbols the frame-invariant proper-force F_o ≡ mα = γ(qv×B) = γdp/dt = γF, where α is the proper-acceleration (were the force unbalanced) felt by the moving-charge, relativistic momentum p ≡ mw in terms of synchrony-free proper-velocity w ≡ dx/dτ = γv where τ is frame-invariant proper-time on the clocks of the moving-charge, Lorentz-factor γ ≡ dt/dτ, coordinate-velocity v ≡ dx/dt, and frame-variant force F ≡ dp/dt (if unbalanced) is the force observed in the neutral-wire frame. Of course x and t are 3-vector position and time (respectively) in the local map-frame, m and q are our moving-ion's rest-mass and charge, while B and E are local magnetic and electric fields.

Note also that:

(i) we are not allowing the force on the moving-charge to change its constant-velocity trajectory e.g. as though it is being counteracted by thrust downward from an "ion-engine" (pun intended),

(ii) we are not showing the magnetic field from the current in the panel at right because it has no effect when the moving charge is stationary, plus

(iii) we are ignoring E-field distortion-effects due to high-speed motion in the wire, if indeed they are an issue for the case of a current-carrying wire.