page_title: Maxwell–Lodge effect
section_title: Description
hierarchical_section_title: Maxwell–Lodge effect / Description
context_page_description: The Maxwell-Lodge effect is a phenomenon of elctromagnetic induction in which an electric charge, near a solenoid in which current changes slowly, feels an electromotive force even if the magnetic field is practically static inside and null outside. It can be considered a classical analogue of the quantum mechanical Aharonov–Bohm effect, where instead the field is exactly static inside and null outside.
  The term appeared in the scientific literature in a 2008 article, referring to an article of 1889 by physicist Oliver Lodge.
context_section_description: Consider an infinite solenoid (ideal solenoid) with n turns per length unit, through which a current  flows. The magnetic field inside the solenoid is,
         (1)
  while the field outside the solenoid is null.
  From the second and third Maxwell's equations,
  and from definitions of magnetic potential and electric potential stems:
  that without electric charges reduces to
         (2)
  Resuming the original definition of Maxwell on the potential vector, according to which is a vector that its circuitation along a closed curve is equal to the flow of   through the surface having the above curve as its edge, i.e. 
  ,
  we can calculate the induced e.m.f., as Lodge did in his 1889 article, considering  the closed line around the solenoid, or convenience a circumference, and  the surface having  as border. Assuming  the radius of the solenoid and  the radius of , the surface crossing it is subjected to a magnetic flux  which is equal to circuitation : . From that stems
  .
  From (2) we have that the e.m.f. is null for  constant, which means, due to (1), at constant current.
  On the other hand, if the current changes,  must also change, producing electromagnetic waves in the surrounding space that can induce an e.m.f. outside the solenoid.
  But if the current changes very slowly, one finds oneself in an almost stationary situation in which the radiative effects are negligible and therefore, excluding , the only possible cause of the e.m.f. is . 
  It is possible to make calculations without referring to the field . Indeed, in the framework of Maxwell equations as written above:
  ,
  being  negligible outside the solenoid. Thus
  .
  This doesn't avoid the problem that  is practically null in places where the e.m.f. manifests itself.
caption: Solenoid and B field with the flow through a surface S of base l