For a infinite, neutral wire in z direction, with steady current i. a charge, q, is placed a distance s below the wire with velocity v parallel to the wire.
a. calculate the lorentz-covariant force law.
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b. transform to new frame where particle is at rest. and compute the force on the particle in the boosted frame.

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this could be newton#'s law of cooling. the cooling curve is exponential

the answer is the first electromagnet (a). this is because it has more coils around the ferromagnetic material. increasing the number of coils increases the surface area in which the metal interacts with the current in the coil. this enables more atoms in the metal to become aligned hence becoming more magnetized.