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A Derivation of Faraday's law from Coulomb's Law and Relativity

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  A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-14 22:45 +0000
    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-15 09:20 +0100
      Re: A Derivation of Faraday's law from Coulomb's Law and Relativity x <x@x.net> - 2026-01-15 13:12 -0800
      Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-15 22:20 +0000
        Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-16 08:22 +0100
          Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-16 11:50 +0000
            Re: A Derivation of Faraday's law from Coulomb's Law and Relativity "Paul B. Andersen" <relativity@paulba.no> - 2026-01-16 22:18 +0100
              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-17 11:58 +0000
            Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-17 15:13 +0100
              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-17 20:34 +0000
                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-18 05:28 +0100
                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-18 19:11 +0000
                    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-18 23:48 +0100
                    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-18 23:50 +0100
                      Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-19 20:51 +0000
                        Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-19 22:03 +0100
                          Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-20 12:01 +0000
                            Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-20 15:08 +0100
                              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-20 19:43 +0000
              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-18 04:03 +0100
            Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-18 18:46 +0000
              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-20 12:16 +0000
                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-20 13:13 +0000
                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-20 14:39 +0000
                    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-20 15:02 +0000
                    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-20 15:26 +0000
                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-20 19:58 +0000
                    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-20 20:58 +0000
                      Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-20 21:00 +0000
                      Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-21 20:27 +0000
                        Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-21 21:20 +0000
                          Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-22 13:35 +0000
                            Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-23 13:19 +0100
                              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-24 06:37 +0100
                              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-24 17:37 +0100
                              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-26 11:03 +0000
                                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-26 15:12 +0100
                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-22 13:07 +0000
                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity John Hasler <john@sugarbit.com> - 2026-01-20 08:05 -0600
                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-20 19:39 +0000
                    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-21 02:12 +0100
                      Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-21 20:21 +0000
                        Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-27 22:37 +0100
                          Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-27 22:50 +0100
                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-20 17:11 +0100
                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-20 20:02 +0000
                    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-21 08:28 +0100
                      Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-21 20:09 +0000
                        Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-27 22:45 +0100
                          Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-28 19:43 +0000
                            Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-29 04:19 +0100
                              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-29 21:45 +0000
                                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-29 22:08 +0000
                                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-30 18:32 +0000
                                    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-30 19:04 +0000
                                      Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-30 19:47 +0000
                                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity John Hasler <john@sugarbit.com> - 2026-01-29 17:11 -0600
                                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-30 01:19 +0100
                                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-30 18:09 +0000
                                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-30 01:18 +0100
                                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-30 18:26 +0000
                              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-30 14:09 +0100
                            Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-29 11:54 +0000
                              Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-29 21:51 +0000
                                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-29 22:13 +0000
                                Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-30 01:20 +0100
                                  Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-30 18:07 +0000
                                    Re: A Derivation of Faraday's law from Coulomb's Law and Relativity ram@zedat.fu-berlin.de (Stefan Ram) - 2026-01-30 18:17 +0000
                                      Re: A Derivation of Faraday's law from Coulomb's Law and Relativity Kuan Peng <titang78@gmail.com> - 2026-01-30 18:33 +0000

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#894931

Kuan Peng <titang78@gmail.com> 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.

  Lenz's Law (1834):

|An induced current is always in such a direction as to oppose
|the motion or change causing it.

  . Web:

|Lenz's Law ensures that induced currents flow in a direction
|that opposes the change in magnetic flux, promoting energy
|conservation.

  .

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#894938

Le 18/01/2026 à 19:46, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
> Kuan Peng <titang78@gmail.com> 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 .

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

And there is no law in electromagnetism that defines a “field that 
pushes back ” . So, we need to correct Faraday’s law or create a new 
law to define the “field that pushes back ”  

Kuan Peng  

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#894939

Kuan Peng <titang78@gmail.com> wrote or quoted:
>And there is no law in electromagnetism that defines a “field that 
>pushes back ” . So, we need to correct Faraday’s law or create a new 
>law to define the “field that pushes back ”  

  The back-reaction of the second coil on the power follows from Ohm's
  law and Maxwell's equations, but not from Faraday's law alone.

  The second coil has an emf acting on it by Faraday's law. 
  By Ohm's law a current appears in the second coil. According
  to the Ampere-Maxwell equation (part of Maxwell's equations)

rot B = μ_0 J + μ_0 e_0 dE/dt, 

  a field arises from this current (here you can use the 
  magnetostatic approximation

rot B = μ_0 J

  ). J here is the current in the second coil. 

  A portion of this field creates a magnetic flux through
  the first coil, which leads to an EMF in the first coil
  by Faraday's law, increasing the load on the power supply.

  Unicode:

𝛁 × 𝐁 = μ₀ 𝐉 + μ₀ ϵ₀ ∂𝐄/∂t

𝛁 × 𝐁 = μ₀ 𝐉  (approximation)

  Summary of some laws:

  Faraday's law states that a time-varying magnetic field induces
  a circling electric field.

  The Ampère-Maxwell law states that currents and time-varying
  electric fields produce circling magnetic fields.

  Ohm's law states that the current density through a conductor
  is proportional to the electric field.

  Lenz's law states that an induced current flows in a direction
  such that the magnetic field it produces opposes the change
  in magnetic flux that induced it. It can be derived from
  Faraday's law, the Ampère–Maxwell law, Ohm's law, and the
  conservation of energy (or the Lorentz force law).

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#894942

ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:
>The back-reaction of the second coil on the power follows from Ohm's
>law and Maxwell's equations, but not from Faraday's law alone.

  It may be instructive to say that Maxwell's laws

Gauss's law (electricity): div E = rho / epsilon_0

Gauss's law (magnetism): div B = 0

Faraday's law: curl E = - dB/dt

Ampere-Maxwell law: curl B = mu_0 J + mu_0 epsilon_0 dE/dt

  can be written as just

d F = 0, d *F = J

  after E and B are united to the two-form F (d being the exterior
  differential, * the Hodge dual, and J the current 3-Form).

  Even if one is not familiar with the exterior differential, this
  show the unity of Maxwell's equations. They are just one single
  law. So, this might make it clear that while "Faray's law" was
  found before Maxwell's equations historically, one cannot actually
  just split Faraday's law away and consider it in isolation always.

  (I'm kinda tempted to define an operator "D" as the pairing
  (d,d*) and a pairing j as (0,J) and then write "DF=j". Well,
  you can see it this way, but that's my non-standard notation.)

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#894943

ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:
>Even if one is not familiar with the exterior differential, this
>show the unity of Maxwell's equations. They are just one single
>law. So, this might make it clear that while "Faray's law" was
>found before Maxwell's equations historically, one cannot actually
>just split Faraday's law away and consider it in isolation always.

  Sorry, there were some typos above!

  Michael Faraday (1791 - 1867), born to a blacksmith in a London
  slum, devoured science from books he bound as a teen apprentice,
  meticulously transcribing Humphry Davy's Royal Institution lectures
  and boldly presenting himself as a lab assistant in 1813 - Davy,
  awed by the young autodidact's zeal, hired him despite his wife
  Jane's persistent snobbery and mistreatment of the lowly upstart.

  Faraday's genius erupted in 1831 with his seminal discovery of
  electromagnetic induction - now immortalized as Faraday's law
  (o-int E dl = -d(Phi_B)/dt) - achieved by rigging coils around
  an iron ring until a twitching galvanometer revealed that
  a changing magnetic field births an electric current, powering 
  the first dynamo prototype where a spinning copper disc atop
  a horseshoe magnet generated ceaseless electricity, all while
  he dazzled Friday crowds with fireworks-like demos, liquefied
  chlorine for fridges, and twisted light rays with magnets [1]
  in notebook eureka moments, twice rejecting the Royal Society
  presidency amid lifelong humility.

  [1] The Faraday effect causes a polarization rotation which is
  proportional to the projection of the magnetic field along the
  direction of the light propagation.

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#894944

ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:
>d F = 0, d *F = J

  I looked it up in Thirring (A Course in Mathematical Physics II)
  and he actually writes, "d *F = - *J".

  There is the concept of the codifferential delta, that allows
  to write this as, "delta F = J", in Unicode, "δF=J".

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#894948

Le 20/01/2026 à 14:13, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>   The back-reaction of the second coil on the power follows from Ohm's
>   law and Maxwell's equations, but not from Faraday's law alone.
> 
>   The second coil has an emf acting on it by Faraday's law. 
>   By Ohm's law a current appears in the second coil. According
>   to the Ampere-Maxwell equation (part of Maxwell's equations)
The second coil has an emf acting on it by Faraday's law. This EMF is 
constant. So, the current in the second coil is constant. 

>   a field arises from this current 
The field from this current is constant.
> 
>   A portion of this field creates a magnetic flux through the first coil,
This magnetic flux is constant because the current in the second coil is 
constant. 
 
>   which leads to an EMF in the first coil by Faraday's law, 
Constant magnetic flux does not change, so " which leads to an EMF " which 
is zero in the first coil. 

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#894950

Kuan Peng <titang78@gmail.com> wrote or quoted:
>Le 20/01/2026 à 14:13, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>The second coil has an emf acting on it by Faraday's law. This EMF is 
>constant. So, the current in the second coil is constant. 

  Ok.

>>a field arises from this current 
>The field from this current is constant.

  Ok.

>>A portion of this field creates a magnetic flux through the first coil,
>This magnetic flux is constant because the current in the second coil is 
>constant. 

  Ok.

>>which leads to an EMF in the first coil by Faraday's law, 
>Constant magnetic flux does not change, so " which leads to an EMF " which 
>is zero in the first coil. 

  I see that I have not addressed this point before. Let me give
  it a try!

  The (increasing) current I1(t) in the first coil creates a flux 

u11(t) = L1 I1(t)

  through the first loop, where L1 is the self-inductance of
  the first loop (by the definition of inductance). The "(t)"
  is intended to indicate the time dependency.

  The (constant) current I2 in the second coil creates a flux 

u12 = M I2 

  through the first loop where M is the mutual inductance of the
  loops (by the definition of the mutual inductance).

  The total flux through the first loop is 

u1(t) = u11(t) + u12 
      = L1 I1(t) + M I2. 

  The sign of I2 is opposite that of I1 by Lenz's law. 

  So one can write: u1(t) = L1 |I1(t)| - M |I2|. 

  The flux u1(t) is reduced by M |I2| even if I2 is constant.

  Thus, to get the same flux as without the other coil, |I1(t)|
  must be greater, which requires more energy than without the
  other coil.

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#894951

ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:
>through the first loop, where L1 is the self-inductance of

  PS: "loop" is intended to have the same meaning as "coil" here.

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#894956

Le 20/01/2026 à 21:58, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>   Thus, to get the same flux as without the other coil, |I1(t)|
>   must be greater, which requires more energy than without the
>   other coil.
Great. But Faraday’s law does not specify how much more energy than 
without the other coil. This is the missing term of Faraday’s law.  

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#894957

Kuan Peng <titang78@gmail.com> wrote or quoted:
>Great. But Faraday’s law does not specify how much more energy than 
>without the other coil. This is the missing term of Faraday’s law.  

  Faraday's law states that a time-varying magnetic field induces
  a circling electric field.

  It does not give that energy. 

  But that does not mean that terms need to be added to Faraday's
  law because Faraday's law is not meant to describe everything
  in the world when it is taken in isolation.

  That energy? It can be calculated using a combination of several laws.

  Faraday's law gives the induced emf E2(t) in the second loop from the
  time rate of change of magnetic flux due to the first coil's current.

  E2(t) = - M * dI1/dt

  where M is the mutual inductance and I1(t) is the current in the
  first coil.

  With the loop 2 resistance R known, Ohm's law gives the induced
  current

  I2(t) = E2(t) / R.

  The power dissipated as heat in the second loop is then 

  P2(t) = E2(t) * I2(t) = [E2(t)]^2 / R

  The extra energy delivered to the second loop over some time
  interval [t0, t1] is

  W2 = integral from t0 to t1 of P2(t) dt
     = integral from t0 to t1 of [E2(t)]^2 / R dt

  Thus, Faraday's law provides E2(t); combined with the known
  resistance R (and, if needed, the mutual inductance M and
  primary current I1(t)), it determines the additional power
  and energy absorbed in the second coil, without adding any
  new term to Faraday's law itself.

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#894961

Le 21/01/2026 à 22:20, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
> Kuan Peng <titang78@gmail.com> wrote or quoted:
>>Great. But Faraday’s law does not specify how much more energy than 
>>without the other coil. This is the missing term of Faraday’s law.  
> 
>   Faraday's law states that a time-varying magnetic field induces
>   a circling electric field.
> 
>   It does not give that energy. 
> 
>   But that does not mean that terms need to be added to Faraday's
>   law because Faraday's law is not meant to describe everything
>   in the world when it is taken in isolation.
> 
>   That energy? It can be calculated using a combination of several laws.
> 
>   Faraday's law gives the induced emf E2(t) in the second loop from the
>   time rate of change of magnetic flux due to the first coil's current.
> 
>   E2(t) = - M * dI1/dt
> 
>   where M is the mutual inductance and I1(t) is the current in the
>   first coil.
> 
>   With the loop 2 resistance R known, Ohm's law gives the induced
>   current
> 
I see. Faraday’s law is a tool. As a tool, it does not have to respect 
the law of conservation of energy 
Poynting’ theorem is a tool and does not have to respect the law of 
conservation of energy
Ohm's law is a tool and does not have to respect the law of conservation 
of energy
We have to combine several laws to fabricate a global solution that 
respects the law of conservation of energy 

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#894963

Kuan Peng wrote:
> Le 21/01/2026 à 22:20, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>> Kuan Peng <titang78@gmail.com> wrote or quoted:
>>> Great. But Faraday’s law does not specify how much more energy than 
>>> without the other coil.

Gibberish.

>> This is the missing term of Faraday’s law.  
>>
>>   Faraday's law states that a time-varying magnetic field induces
>>   a circling electric field.
>>
>>   It does not give that energy. 
>>
>>   But that does not mean that terms need to be added to Faraday's
>>   law because Faraday's law is not meant to describe everything
>>   in the world when it is taken in isolation.
>>
>>   That energy? It can be calculated using a combination of several laws.
>>
>>   Faraday's law gives the induced emf E2(t) in the second loop from the
>>   time rate of change of magnetic flux due to the first coil's current.
>>
>>   E2(t) = - M * dI1/dt
>>
>>   where M is the mutual inductance and I1(t) is the current in the
>>   first coil.
>>
>>   With the loop 2 resistance R known, Ohm's law gives the induced
>>   current
>
> I see. Faraday’s law is a tool.

No, Faraday's law _of induction_ (how many more times do I have to tell
you?) is an empirically confirmed physical law.

AISB, you could not read this if it were fundamentally wrong: most electric
appliances, certainly electronic devices, include transformers (or require
power adapters which include them to transform high voltage to low voltage)
which are working based on that law:

<https://en.wikipedia.org/wiki/Transformer>

> As a tool, it does not have to respect the law of conservation of energy 

No, you simply have no clue what you are talking about.

> Poynting’ theorem is a tool and does not have to respect the law of 
> conservation of energy

No, you simply have no clue what you are talking about.

> Ohm's law is a tool and does not have to respect the law of conservation 
> of energy

No, you simply have no clue what you are talking about.

> We have to combine several laws to fabricate a global solution that 
> respects the law of conservation of energy 

No; as I have proved to you in the very beginning, when we calculate the
change of the energy density of the electromagnetic field with respect to
time, the continuity equation for (classical) electrodynamics including
the Poynting vector and the work done by the elctromagnetic field, results
*naturally*.

What I have not shown is how to calculate the energy density itself because
it is a rather lengthy calculation.  But it can be shown by integrating the
Lorentz force

  F = q (E + V × B)

over the path of a charged particle, which in turn can be derived from the
special-relativistic (Lorentz-covariant) Lagrangian for a charged particle
coupled to the electromagnetic field (given by the Maxwell tensor with
components F_ab = ∂_a A_b − ∂_b A_a, where A_a = [−ϕ/c, A] is the
four-potential, where ϕ is the electric potential in E = −∇ϕ, and A is the
magnetic vector potential in B = ∇ × A).

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

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#894964

Thomas 'PointedEars' Lahn wrote:
> What I have not shown is how to calculate the energy density itself because
> it is a rather lengthy calculation.

By that I meant how to _derive the formula_ for the energy density of the EM
field in Gaussian units,

  u(X, t) = 1/(8π) [E²(X, t) + B²(X, T)].

Once you have that formula, its value is rather trivial to calculate, of course.

> But it can be shown by integrating the Lorentz force
> 
>   F = q (E + V × B)
> 
> over the path of a charged particle,

Hint: The absolute value of the work done by the electromagnetic field on a
charged particle,

  W = ∫_P dS ⋅ F = ∫_P dS ⋅ q (E + V × B)

is equal to the difference between the energy that was stored in the field
before it did that work and after that.  Notice that if dS is an
infinitesimal line element of the particle's spatial path P, then dS || V
which means that the magnetic field does not do work on the particle:
dS ⋅ q (V × B) = 0.  (But that does not mean that no energy is stored in it.)

> which in turn can be derived from the special-relativistic (Lorentz-covariant)
> Lagrangian for a charged particle coupled to the electromagnetic field (given
> by the Maxwell tensor with components F_ab = ∂_a A_b − ∂_b A_a, where
> A_a = [−ϕ/c, A] is the four-potential, where ϕ is the electric potential in
> E = −∇ϕ, and A is the magnetic vector potential in B = ∇ × A).

Hint: The total action is the sum of the action for a free particle,

  S_0[x] = -m c^2 ∫_W dτ,

where τ is proper time, and the action for the interaction with the EM field,

  S_i[x; A] = q ∫_W dτ A_a dx^a/dτ,

where W is a section of the (timelike) worldline of the particle.  As usual,
one can find the Euler--Lagrange equations by calculating the variation of
the total action, here

  S[x; A] = S_0[x] + S_I[x; A].

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

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#894965

[Supersedes because of too many typos]

Thomas 'PointedEars' Lahn wrote:
> What I have not shown is how to calculate the energy density itself because
> it is a rather lengthy calculation.

By that I meant how to _derive the formula_ for the energy density of the EM
field in Gaussian units,

  u(X, t) = 1/(8π) [E²(X, t) + B²(X, t)].

Once you have that formula, its value is rather trivial to calculate, of course.

> But it can be shown by integrating the Lorentz force
> 
>   F = q (E + V × B)
> 
> over the path of a charged particle,

Hint: The absolute value of the work done by the electromagnetic field on a
charged particle,

  W = ∫_P dS ⋅ F = ∫_P dS ⋅ q (E + V × B)

is equal to the difference between the energy that was stored in the field
before it did that work and after that.  Notice that if dS is an
infinitesimal line element of the particle's spatial path P, then dS || V
which means that the magnetic field does not do work on the particle:
dS ⋅ q (V × B) = 0.  (But that does not mean that no energy is stored in it.)

> which in turn can be derived from the special-relativistic (Lorentz-covariant)
> Lagrangian for a charged particle coupled to the electromagnetic field (given
> by the Maxwell tensor with components F_ab = ∂_a A_b − ∂_b A_a, where
> A_a = [−ϕ/c, A] is the four-potential, where ϕ is the electric potential in
> E = −∇ϕ, and A is the magnetic vector potential in B = ∇ × A).

Hint: The total action is the sum of the action for a free particle,

  S_0[x] = -m c^2 ∫_W dτ,

where W is a section of the (timelike) worldline of the particle, τ is
proper time, and the action for the interaction with the EM field,

  S_I[x; A] = q ∫_W dτ A_a dx^a/dτ.

As usual, one can find the Euler--Lagrange equations by calculating the
variation of the total action, here

  S[x; A] = S_0[x] + S_I[x; A].

Because the variation is a *linear* differential operation, and the
Euler--Lagrange equations are *linear* differential equations, knowing the
free relativistic Lagrangian, it suffices to vary S_I[x; A] to obtain the
equations of motion for the interaction and add the non-interacting ones to
obtain the Lorentz force equations.

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

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#894966

Le 23/01/2026 à 13:19, Thomas 'PointedEars' Lahn a écrit :
> Kuan Peng wrote:
>> Le 21/01/2026 à 22:20, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>>> Kuan Peng <titang78@gmail.com> wrote or quoted:
>>>> Great. But Faraday’s law does not specify how much more energy than 
>>>> without the other coil.
> 
> Gibberish.
> 
>>> This is the missing term of Faraday’s law.  
>>>
>>>   Faraday's law states that a time-varying magnetic field induces
>>>   a circling electric field.
>>>
>>>   It does not give that energy. 
>>>
>>>   But that does not mean that terms need to be added to Faraday's
>>>   law because Faraday's law is not meant to describe everything
>>>   in the world when it is taken in isolation.
>>>
>>>   That energy? It can be calculated using a combination of several laws.
>>>
>>>   Faraday's law gives the induced emf E2(t) in the second loop from the
>>>   time rate of change of magnetic flux due to the first coil's current.
>>>
>>>   E2(t) = - M * dI1/dt
>>>
>>>   where M is the mutual inductance and I1(t) is the current in the
>>>   first coil.
>>>
>>>   With the loop 2 resistance R known, Ohm's law gives the induced
>>>   current
>>
>> I see. Faraday’s law is a tool.
> 
> No, Faraday's law _of induction_ (how many more times do I have to tell
> you?) is an empirically confirmed physical law.
> 
> AISB, you could not read this if it were fundamentally wrong: most electric
> appliances, certainly electronic devices, include transformers (or require
> power adapters which include them to transform high voltage to low voltage)
> which are working based on that law:
> 
> <https://en.wikipedia.org/wiki/Transformer>
> 
>> As a tool, it does not have to respect the law of conservation of energy 
> 
> No, you simply have no clue what you are talking about.
> 
>> Poynting’ theorem is a tool and does not have to respect the law of 
>> conservation of energy
> 
> No, you simply have no clue what you are talking about.
> 
>> Ohm's law is a tool and does not have to respect the law of conservation 
>> of energy
> 
> No, you simply have no clue what you are talking about.
> 
>> We have to combine several laws to fabricate a global solution that 
>> respects the law of conservation of energy 
> 
> No; as I have proved to you in the very beginning, when we calculate the
> change of the energy density of the electromagnetic field with respect to
> time, the continuity equation for (classical) electrodynamics including
> the Poynting vector and the work done by the elctromagnetic field, results
> *naturally*.
> 
> What I have not shown is how to calculate the energy density itself because
> it is a rather lengthy calculation.  But it can be shown by integrating the
> Lorentz force
> 
>   F = q (E + V × B)
> 
> over the path of a charged particle, which in turn can be derived from the
> special-relativistic (Lorentz-covariant) Lagrangian for a charged particle
> coupled to the electromagnetic field (given by the Maxwell tensor with
> components F_ab = ∂_a A_b − ∂_b A_a, where A_a = [−ϕ/c, A] is the
> four-potential, where ϕ is the electric potential in E = −∇ϕ, and A is the
> magnetic vector potential in B = ∇ × A).
Faraday’s law is the same for all. I see now the difference of the 
understanding between you and me.

Thank you.

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#894967

Please trim your quotes to the relevant minimum.

Kuan Peng wrote:
> Faraday’s law is the same for all.
What do you mean by that?

> I see now the difference of the understanding between you and me.

I am not sure that you have understood me properly; anyhow:

> Thank you.

You are welcome.

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

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#894959

ram@zedat.fu-berlin.de (Stefan Ram) wrote or quoted:
>rot B = μ_0 J + μ_0 e_0 dE/dt, 

  Oops! "rot" should be "curl" above.

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#894940

Kuan Peng writes:
> 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

Coil B is a coil with current in it.  Faraday's law predicts that
it will generate a field which opposes that generated by coil A.
-- 
John Hasler 
john@sugarbit.com
Dancing Horse Hill
Elmwood, WI USA

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#894946

Le 20/01/2026 à 15:08, John Hasler a écrit :
> Kuan Peng writes:
>> 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
> 
> Coil B is a coil with current in it.  Faraday's law predicts that
> it will generate a field which opposes that generated by coil A.

What if the current in B is constant?

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