From: PointedEars@web.de   
      
   Kuan Peng wrote:   
   > Le 18/01/2026 à 19:46, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :   
   >> Kuan Peng  wrote or quoted:   
   >>> Since the energy consumption in coil A is zero, A does not transfer any   
   >>> energy to coil B.   
   >>   
   >>   Electromagnetism, especially the part with coils, isn't exactly   
   >>   my strong suit! But if I had to take a stab at it, I'd say the   
   >>   current in coil B creates a field that pushes back against the   
   >>   change in current in coil A (effectively Lenz's law). So you   
   >>   end up having to put in extra energy to keep the current rising   
   >>   linearly, and that's the energy that gets dissipated.   
   >   
   > Yes. you are absolutely right. This is how energy gets balanced in coils A   
   > and B in real experiment .   
      
   There is no "balancing of energy".   
      
   > However, Faraday’s law does not define :   
   >>   the current in coil B creates a field that pushes back against the   
   >>   change in current in coil A   
      
   Because that is not what that law is about.  However, with the   
   Ampère--Maxwell Law (Maxwell's refinement of Ampère's Circuital Law¹), you   
   can see that the induced electric field (which produces the current) induces   
   "a(nother)" magnetic field².   
      
   We begin with Faraday’s law _of induction_ (AISB, there are _several_   
   "Faraday's laws").  In differential form and SI units, it is   
      
     ∇ × E = −∂B∕∂t.    (1)   
      
   The Ampère--Maxwell Law is in differential form and SI units   
      
     ∇ × B = μ₀ J + (1/c²) ∂E∕∂t,    (2)   
      
   where the capital letters mean vector fields.   
      
   But when we are thinking about "a (magnetic) (B) field that pushes back   
   against the change in current in coil A", that field is *not the same*   
   B field as in eq. (1).  So we should label it differently, for example   
      
     ∇ × B' = μ₀ J + (1/c²) ∂E∕∂t.    (3)   
      
     [I am not sure yet if this E is the same E as in eq. (1), or the E in   
      eq. (1) only produces the J here.  If it is not the same E, it should   
      be labeled differently as well, e.g. "E'".]   
      
   [Without proof yet:]   
      
   Eventually, the resulting magnetic field is the superposition of B and B';   
   the polarity of B' is opposite that of B, so it weakens the B field, so to   
   speak, thus its change reduces the electric current produced by the change   
   of the B-field.²   
      
   > And there is no law in electromagnetism that defines a “field that   
   > pushes back ” .   
      
   There is; it is called Lenz's Law:   
      
      
      
   It can probably be derived from the equations above; maybe I will do it when   
   I have more time.   
      
   > So, we need to correct Faraday’s law or create a new   
   > law to define the “field that pushes back ”   
      
   /Ex falso quodlibet./   
      
   ___   
   ¹  Maxwell discovered that the total electric charge would only be conserved   
      if he modified Ampère's Circuital Law by adding a -- what he called --   
      "displacement current".  [Since I have done the derivation again anyway   
      in an attempt to derive Lenz's Law, I might as well post it :'-)]   
      
      If we calculate the divergence of the left-hand side and right-hand side   
      of eq. (2), we obtain   
      
             ∇ ⋅ (∇ × B') = ∇ ⋅ (μ₀ J + ε₀ μ₀ ∂E∕∂t)   
        <==>             0 = μ₀ (∇ ⋅ J) + ε₀ μ₀ ∂∕∂t (∇ ⋅   
   E)   
      
      because the divergence of a curl field is zero ("a field with closed   
      field lines has no sources"), and partial derivatives of twice   
      differentiable functions commute (Schwarz--Clairaut Theorem).  But we   
      also have Gauss' Law:   
      
        ∇ ⋅ E = ρ/ε₀,   
      
      where ρ is the electric charge density.  So   
      
                 0 = μ₀ (∇ ⋅ J) + ε₀/ε₀ μ₀ ∂ρ∕∂t   
        <==>     0 = ∇ ⋅ J + ∂ρ∕∂t.   
        <==> ∇ ⋅ J = -∂ρ∕∂t.   
      
      This is the (non-relativistic) *continuity equation* for classical   
      electrodynamics.  In words, it means: For an electric current to flow out   
      of a volume of space, the electric charge density in that volume must   
      decrease.   
      
      In other words, the total electric charge is conserved: When the electric   
      charge decreases in one volume, it must increase in another (the   
      adjacent) one.  Or, if the charge density in a volume remains constant,   
      either there is no electric current passing through that volume, or as   
      much electric charge flows into it as out of it.   
      
      In yet other words, it is not possible to produce electric charge out   
      of nowhere (you have to take away the required amount of opposite   
      charge, which is what we actually mean by "charging"), or to destroy it   
      without neutralizing it.   
      
   ²  There is only one magnetic field, actually only one electromagnetic   
      field.  But it is useful to speak of the contributions to either by   
      different processes as different fields.  This is allowed by the   
      principle of superposition which applies here because Maxwell's   
      equations are *linear* differential equations, and so the sum of   
      two solutions is also a solution.  For example, if   
      
        ∇ × E_1 = −∂∕∂t B_1   
        ∇ × E_2 = −∂∕∂t B_2,   
      
      i.e. E_1 and B_1, and E_2 and B_2, are pairwise solutions of this   
      equation, then   
      
             ∇ × E_1 + ∇ × E_2 = −∂∕∂t B_1 − ∂∕∂t B_2   
        <==> ∇ × (E_1 + E_2)    = −∂∕∂t (B_1 + B_2),   
      
      so E_1 + E_2 and B_1 + B_2 are solutions as well. ∎   
   --   
   PointedEars   
      
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