section_path: Relation to other physical theories/Electromagnetism
section_name: Electromagnetism
paragraph_text: Electromagnetism
  
  Newton's three laws can be applied to phenomena involving electricity and magnetism, though subtleties and caveats exist.
  
  Coulomb's law for the electric force between two stationary, electrically charged bodies has much the same mathematical form as Newton's law of universal gravitation: the force is proportional to the product of the charges, inversely proportional to the square of the distance between them, and directed along the straight line between them. The Coulomb force that a charge q 1 {\displaystyle q_{1}} exerts upon a charge q 2 {\displaystyle q_{2}} is equal in magnitude to the force that q 2 {\displaystyle q_{2}} exerts upon q 1 {\displaystyle q_{1}}, and it points in the exact opposite direction. Coulomb's law is thus consistent with Newton's third law.
  
  Electromagnetism treats forces as produced by fields acting upon charges. The Lorentz force law provides an expression for the force upon a charged body that can be plugged into Newton's second law in order to calculate its acceleration. According to the Lorentz force law, a charged body in an electric field experiences a force in the direction of that field, a force proportional to its charge q {\displaystyle q} and to the strength of the electric field. In addition, a moving charged body in a magnetic field experiences a force that is also proportional to its charge, in a direction perpendicular to both the field and the body's direction of motion. Using the vector cross product, F = q E + q v × B. {\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B}.}
  
  If the electric field vanishes (E = 0 {\displaystyle \mathbf {E} =0}), then the force will be perpendicular to the charge's motion, just as in the case of uniform circular motion studied above, and the charge will circle (or more generally move in a helix) around the magnetic field lines at the cyclotron frequency ω = q B / m {\displaystyle \omega =qB/m}. Mass spectrometry works by applying electric and/or magnetic fields to moving charges and measuring the resulting acceleration, which by the Lorentz force law yields the mass-to-charge ratio.
  
  Collections of charged bodies do not always obey Newton's third law: there can be a change of one body's momentum without a compensatory change in the momentum of another. The discrepancy is accounted for by momentum carried by the electromagnetic field itself. The momentum per unit volume of the electromagnetic field is proportional to the Poynting vector.
  
  There is subtle conceptual conflict between electromagnetism and Newton's first law: Maxwell's theory of electromagnetism predicts that electromagnetic waves will travel through empty space at a constant, definite speed. Thus, some inertial observers seemingly have a privileged status over the others, namely those who measure the speed of light and find it to be the value predicted by the Maxwell equations. In other words, light provides an absolute standard for speed, yet the principle of inertia holds that there should be no such standard. This tension is resolved in the theory of special relativity, which revises the notions of space and time in such a way that all inertial observers will agree upon the speed of light in vacuum.