To make this explicable: if a conductor moves through a B-field with a gradient ∂ ∂ B z / ∂ ∂ z , along the z-axis with constant velocity v z = ∂ ∂ z / ∂ ∂ t , it follows that in the frame of the conductor ∂ ∂ B z ′ ∂ ∂ t = v z ∂ ∂ B z ∂ ∂ z = − − ( ∇ ∇ × × E ′ ) z = ∂ ∂ E x ′ ∂ ∂ y − − ∂ ∂ E y ′ ∂ ∂ x . It can be seen that this equation is consistent with E ′ = v × × B = v z B x y ^ ^ − − v z B y x ^ ^ , by determining ∂ ∂ E x ′ / ∂ ∂ y and ∂ ∂ E y ′ / ∂ ∂ x from this expression and substituting it in the first expression while using that ∇ ∇ ⋅ ⋅ B = ∂ ∂ B x ∂ ∂ x + ∂ ∂ B y ∂ ∂ y + ∂ ∂ B z ∂ ∂ z = 0. Even in the limit of infinitesimal small gradients ∂ ∂ B z / ∂ ∂ z these relations hold, and therefore the Lorentz force equation is also valid if the magnetic field in the conductor frame is not varying in time. At relativistic velocities a correction factor is needed, see below and Classical electromagnetism and special relativity and Lorentz transformation.