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Modified Josephson Relation

For type II superconductors, Josephson has shown that vortices moving with velocity v_L create an effective electric field E'=-v_L x B. By definition the effective electric field is gradient of the electrochemical potential, what is the quantity corresponding to voltage observed with the use of Ohmic contacts. It relates to the true electric field E via the local chemical potential mu as E'=E - grad(mu)/e. We argue that at low temperatures the true electric field in the bulk can be approximated by a modified Josephson relation E=(v_s-v_L) x B, where v_S is the condensate velocity.