From: PointedEars@web.de   
      
   Kuan Peng wrote:   
   > I have explained this violation of the law of conservation of energy in   
   > the introduction.   
      
   The total energy is conserved in electrodynamics as well, which can be   
   proved:¹   
      
   The energy density of the electromagnetic field is in Gaussian units   
      
     u(X, t) = (1/8π) [|E(X, t)|^2 + |B(X, t)|^2],   
      
   where X is 3-position, and E and B are the electric and magnetic (flux   
   density) field vectors, respectively.  We can also write this   
      
     u(X, t) = (1/8π) [E(X, t) ⋅ E(X, t) + B(X, t) ⋅ B(X, t)],   
      
   where "⋅" indicates the standard real scalar product.   
      
   Taking the partial derivative with respect to time, we find by the product rule   
      
     ∂_t u(X, t) = (1/8π) [∂_t E(X, t) ⋅ E(X, t) + E(X, t) ⋅ ∂_t E(X,   
   t)   
                         + ∂_t B(X, t) ⋅ B(X, t) + B(X, t) ⋅ ∂_t B(X, t)]   
      
                 = (1/8π) [2 E(X, t) ⋅ ∂_t E(X, t) + 2 B(X, t) ⋅ ∂_t   
   B(X, t)]   
                 = (1/4π) (E ⋅ ∂_t E + B ⋅ ∂_t B),   
      
   where for simplicity in the last step I have not written the coordinate   
   dependence of the fields anymore.   
      
   Using two of Maxwell's equations in Gaussian units,   
      
     ∇ × E = -(1/c) ∂_t B              [Faraday's Law of Induction]   
     ∇ × B = (4π/c) J + (1/c) ∂_t E    [Ampère--Maxwell Circuital Law],   
      
   where J is the electric current density, we can write   
      
     ∂_t u = (1/4π) {E ⋅ [c ∇ × B - 4π J] - c B ⋅ (∇ × E)}   
           = (1/4π) {c E ⋅ (∇ × B) - 4π E ⋅ J - c B ⋅ (∇ × E)}   
           = (c/4π) {E ⋅ (∇ × B) - B ⋅ (∇ × E)} - E ⋅ J.   
      
   We can use the product rule   
      
     ∇ ⋅ (A × B) = B ⋅ (∇ × A) - A ⋅ (∇ × B)   
      
   to write   
      
     ∂_t u(X, t) = -(c/4π) ∇ ⋅ (E × B) - E ⋅ J.   
      
   one introduces the Poynting vector   
      
               S = (c/4π) (E × B),   
      
   to write the continuity equation   
      
     ∂_t u(X, t) + ∇ ⋅ S + J ⋅ E = 0   
      
   ∇ ⋅ S is basically the energy transported away by electromagnetic waves   
   (which, following vector calculus, propagate in a direction perpendicular   
   to the electric and magnetic field lines).   
      
   J ⋅ E is the work done by the electromagnetic field (actually, only the   
   electric field) per unit volume and unit time:   
      
   The electromagnetic force is given by   
      
     F = q [E + V × B),   
      
   where q is the electric charge of a particle that is moving in an   
   electromagnetic field with the velocity V(R) = dR/dt.  The work done on it   
   by the EM field along a curve C is   
      
     W = ∫_C dR ⋅ F(R)   
       = ∫ dt dR/dt ⋅ F(t)   
       = ∫ dt V(t) ⋅ q [E + V× B)   
       = ∫ dt q [V ⋅ E + V ⋅ (V × B)]   
       = ∫ dt q V ⋅ E,   
      
   [V × B is perpendicular to V, so V ⋅ (V × B) = 0], so the work done per   
   unit   
   time is   
      
     dW/dt = q V ⋅ E = q dR/dt ⋅ E = dq/dt R ⋅ E = I ⋅ E,   
      
   where I is the directed electric current.  The work done by the   
   electromagnetic field per unit volume and unit time is   
      
     dw/dt = J ⋅ E. ∎   
      
   ___   
   ¹ Based on lecture notes for the course "Classical Electrodynamics" (autumn   
     semester 2023) by Uwe-Jens Wiese, University of Bern   
      
   > To illustrate this, consider the following experimental setup: suppose two   
   > coils, A and B, are positioned side by side, with coil B connected to a   
   > resistor R, as shown in Figure 1.   
   >   
   > Let the current in coil A, denoted as Ia, vary as follows: Ia increases   
   > linearly from zero to Imax, then decreases linearly back to zero.   
      
   For the current to *change*, work has to be done, thus energy is transferred   
   to or away from the coil.  Thus the energy stored in the coil is not   
   conserved, but nobody claimed that it would be then; the law is, and this is   
   what happens here, too, that the *total* energy stored in the coil *and* of   
   its environment is conserved.   
      
   > The duration of each phase is Δt. According to Faraday's law, voltages are   
   > induced in coils A and B,   
      
   No, that is NOT what Faraday's law of induction is about.  It states that an   
   electric _current_ is induced by a changing _magnetic field_:   
      
     ∇ × E = -(1/c) ∂B/∂t,   
      
   where E is the electric field vector, B is the magnetic flux density field   
   vector, and t is time.   
      
   But you already presume a changing *current*.   
      
   > The cause of this violation is that Faraday's law predicts zero voltage in   
   > A when the current in coil B is constant.   
      
   No, the reason of this *seeming* violation is that you are confusing   
   yourself by assuming something that Faraday's law is not about, and   
   you have not considered all the facts.   
      
   --   
   PointedEars   
      
   Twitter: @PointedEars2   
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