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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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#894952

Kuan Peng wrote:
> 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?

[Those are vector fields that can depend both on time *and* position.
Therefore, it is important to state precisely with respect to what a field
is constant: Does it not vary over time, or does it perhaps vary over time,
but not in space?]

Then, if the external electric field is constant over time or the external
electric field strength is constantly increasing over time, too, the
resulting magnetic field is constant over time, too, as you can see from the
Ampère--Maxwell Law

  ∇ × B = μ₀ J + (1/c²) ∂E∕∂t.

Take the (partial) derivative with respect to time, then

  ∂/∂t (∇ × B) = μ₀ ∂J/∂t + (1/c²) ∂²E∕∂t².

If I = const., then ||J|| = dI/dA = const., and ∂J/∂t = 0, so

  ∂/∂t (∇ × B) = ∇ × ∂B/∂t = (1/c²) ∂²E∕∂t².

Calculating the surface integral over some cross-sectional area A, one finds
by application of the Kelvin--Stokes Theorem (K/S):

       ∬_A dA ⋅ (∇ × ∂B/∂t) = (1/c²) ∂²E∕∂t² ∬_A dA

   K/S
  <==> ∮_C dL ⋅ ∂B/∂t        = (1/c²) ∂²E∕∂t² ∬_A dA,

where C is the delimiting curve, so eventually

                      ∂B/∂t ~ (1/c²) ∂²E∕∂t² ∬_A dA,

and unless ∂²E∕∂t² ≠ 0, then ∂B/∂t = 0.  ∂²E∕∂t² = 0 if either ∂E∕∂t = 0,
i.e. E = const. (wrt. time), or ∂E∕∂t = const. (wrt. time) ≠ 0. ∎

-- 
PointedEars

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

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#894955

Le 21/01/2026 à 02:12, Thomas 'PointedEars' Lahn a écrit :
> Kuan Peng wrote:
>> 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?
> 
> [Those are vector fields that can depend both on time *and* position.
> Therefore, it is important to state precisely with respect to what a field
> is constant: Does it not vary over time, or does it perhaps vary over time,
> but not in space?]
> 
The current in A varies linearly. The current in B is constant. 
So, the magnetic field of A+B varies linearly.
A and B are both in this magnetic field which varies linearly.
According to Faraday’s law, the induced voltages in A and B are 
proportional to dB/dt which is constant.
The induced voltage and current in A with or without the presence of B is 
the same.
So, the energy dissipation is zero with or without the presence of B.

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#894970

Kuan Peng wrote:
> Le 21/01/2026 à 02:12, Thomas 'PointedEars' Lahn a écrit :
>> Kuan Peng wrote:
>>> 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?
>>
>> [Those are vector fields that can depend both on time *and* position.
>> Therefore, it is important to state precisely with respect to what a field
>> is constant: Does it not vary over time, or does it perhaps vary over time,
>> but not in space?]
>>
> The current in A varies linearly. The current in B is constant. 
> So, the magnetic field of A+B varies linearly.

I wonder how there is a current in (the secondary) coil B at all *before*
electromagnetic induction.  Usually there is not, i.e. the secondary coil
is NOT connected to a voltage source, but to some electric appliance:

<https://en.wikipedia.org/wiki/Electromagnetic_induction#History>

In any case, your third statement is only superficially true because the
change of the current in coil A changes the magnetic field in and around
coil A (which is also the magnetic field in and around around coil B).  The
change of the magnetic field induces another current which counteracts the
one in coil A and perhaps even coil B (Lenz's Law).

That, in a sense, another current is induced by that change follows from
Faraday's law of induction (eq. 2 below) that was used to induce a current
in coil B in the first place; but its direction is not obvious (to me).

In vacuum:

  ∇ × B  = μ₀ (J + ε₀ μ₀ ∂E/∂t) = μ₀ J + (1/c²) ∂E/∂t,    (1)
  ∇ × E' = -∂B'/∂t,                                       (2)

where (IIUC) E' is now the contribution to the electric field that is
induced by the change in the magnetic field B' due to the induced current,
that produces Lenz's opposing current ':-)

> A and B are both in this magnetic field which varies linearly.

It is not, for the reason explained above.

> According to Faraday’s law,

_of induction_ (but I guess we can drop that to simplify this discussion)

> the induced voltages in A and B are proportional to dB/dt

Yes, that follows (sort of) from the differential form above, since the
induced _electric field_ gives rise to a spatial difference of electric
potential, i.e. a voltage.

> which is constant.

It is not.

> The induced voltage and current in A with or without the presence of B is 
> the same.

It is not.

> So, the energy dissipation is zero with or without the presence of B.

/Ex falso quodlibet./

-- 
PointedEars

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

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#894972

Thomas 'PointedEars' Lahn wrote:
> In vacuum:
> 
>   ∇ × B  = μ₀ (J + ε₀ μ₀ ∂E/∂t) = μ₀ J + (1/c²) ∂E/∂t,    (1)

The careful reader has probably noticed it: There is an extra μ₀ between the
parentheses :'-)

>   ∇ × E' = -∂B'/∂t,                                       (2)
> 
> where (IIUC) E' is now the contribution to the electric field that is
> induced by the change in the magnetic field B' due to the induced current,
> that produces Lenz's opposing current ':-)

-- 
PointedEars

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

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#894945

Kuan Peng wrote:
> 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 .

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:

<https://en.wikipedia.org/wiki/Lenz%27s_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

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

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#894949

Le 20/01/2026 à 17:11, Thomas 'PointedEars' Lahn a écrit :
> Kuan Peng wrote:
>> However, Faraday’s law does not define :
> 
> We begin with Faraday’s law _of induction_ (AISB, there are _several_
> "Faraday's laws").  In differential form and SI units, it is
> 
> 
> 
>> And there is no law in electromagnetism that defines a “field that 
>> pushes back ” .
> 
> There is; it is called Lenz's Law:
> 
If at the beginning the current in A is constant, do you think that there 
is a current in the coil B? 

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#894953

Kuan Peng wrote:
> Le 20/01/2026 à 17:11, Thomas 'PointedEars' Lahn a écrit :
>> There is; it is called Lenz's Law:
>
> If at the beginning the current in A is constant,

What exactly do you mean by that?

> do you think that there is a current in the coil B?

Unless coil B is in the vicinity of a voltage source, or if the external
electric field is not constant, then there should not be:

  ∇ × E = −∂B∕∂t,
  ∇ × B = μ₀ J + (1/c²) ∂E∕∂t

and

  J = σ E,

where σ is conductivity, and

  I = ∬ dA ⋅ J,

so there is a current when the magnetic field changes, or there is a
non-zero electric field (that is produced by the voltage source).

But one has to be careful here.  A *measured* current is rarely exactly 0
all the time; for example, there is interference from external source, and
the phenomenon and practical problem of a /Kriechstrom/ (literally: crawling
current).  Experimental evidence that a measured current is not zero is
therefore insufficient to confirm the claim that Faraday's law of induction
would be wrong or needed refinement.  Much of it depends on your
experimental setup.

-- 
PointedEars

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

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#894954

Le 21/01/2026 à 08:29, Thomas 'PointedEars' Lahn a écrit :
>> If at the beginning the current in A is constant,
> 
> What exactly do you mean by that?
> 
>> do you think that there is a current in the coil B?
> 
> Unless coil B is in the vicinity of a voltage source, or if the external
> electric field is not constant, then there should not be:
> 
We discuss the phenomenon in ideal condition with no field other than 
those from A and B.
If you agree that 
1.	When the current in A is constant, the induced voltage and current in B 
are zero. 
Then, 
2.	When the current in B is constant, the induced voltage and current in A 
are zero.

3.	If the current in B is induced by the current in A, the current in B is 
constant. Then, the voltage and current in A induced by the current in B 
are zero, which is our case 2. 

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#894971

Kuan Peng wrote:
> Le 21/01/2026 à 08:29, Thomas 'PointedEars' Lahn a écrit :
>>> If at the beginning the current in A is constant,
>>
>> What exactly do you mean by that?

You have not answered my question.

>>> do you think that there is a current in the coil B?
>>
>> Unless coil B is in the vicinity of a voltage source, or if the external
>> electric field is not constant, then there should not be:
>
> We discuss the phenomenon in ideal condition with no field other than 
> those from A and B.

Technically, the fields are not "from A and B", but exist in all of
spacetime, and just have different values at different locations and
times; but OK.

> If you agree that 
> 1.	When the current in A is constant, the induced voltage and current in B 
> are zero. 

I agree provisionally: *voltages* are _never_ induced, _currents_ are, due
to electric fields.

> Then, 
> 2.	When the current in B is constant, the induced voltage and current in A 
> are zero.

Same as above.

> 3.	If the current in B is induced by the current in A, the current in B is 
> constant.
> Then, the voltage and current in A induced by the current in B 
> are zero, which is our case 2. 

I disagree with both statements because of Lenz's Law.

-- 
PointedEars

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

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#894974

Le 27/01/2026 à 22:45, Thomas 'PointedEars' Lahn a écrit :
> Kuan Peng wrote:
>>> What exactly do you mean by that?
> 
> You have not answered my question.
> 
I mean that the law
>   ∇ × E' = -∂B'/∂t,                                       (2)
is the same for all physicists. We understand it as that the curl of the 
electric field equals minus the rate of change of the magnetic field.

But each physicist attribute it different sense in his head and it is his 
proper interpretation that makes that different physicist when predicting 
the outcome of a same experiment using the same formula ∇ × E' = 
-∂B'/∂t can give different result.

Le 27/01/2026 à 22:37, Thomas 'PointedEars' Lahn a écrit :
> 
> I wonder how there is a current in (the secondary) coil B at all *before*
> electromagnetic induction.  Usually there is not, i.e. the secondary coil
> is NOT connected to a voltage source, but to some electric appliance:
1.	Before the current circulates in coil A, there is not a current in coil 
B .
2.	After the current circulates in coil A, a current is induced in coil B.
3.	Now, we have a current in coil A and a current in coil B.
4.	Because the current in coil A increases linearly, the magnetic field in 
coil B increases linearly and its rate of change is -∂B'/∂t which is 
constant. 
5.	Because-∂B'/∂t is constant, ∇ × E' = -∂B'/∂t is constant. 
So, E' is constant. 
6.	Because E' is constant, the induced voltage in coil B is constant. Let 
Vb be this voltage 
7.	The resistor R is connected to the coil B and the voltage Vb is across 
R, the current in R equals Ib = Vb / R 
8.	Because Vb is constant, Ib = Vb / R is constant.
9.	Now we have the current Ia in coil A and Ib in coil B. Ia generates the 
magnetic field Ba and Ib generates Bb in space.
10.	Now the total magnetic field in space is total B= Ba+ Bb
11.	Because Ba increases linearly and  Bb is constant, total B increases 
linearly with the same rate of change than Ba 
12.	Now the coil A and B are in the magnetic field total B which increases 
linearly and ∂( total B)/∂t is constant   
13.	Because ∇ × E' = -∂( total B)/∂t, E' is constant in the coil A 
and B. So, the induced voltage in the coil A and B, Va and Vb , are both 
constant  
14.	So, when the current Ia in coil A increases linearly, Va>0, Vb>0 when 
the current Ia in coil A decreases linearly, Va<0, Vb<0 .
15.	The dissipation in coil B equals 2Vb*Vb/R*t>0 
16.	The dissipation in coil A equals time integral of (Va-Va) Ia*dt=0   
  

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#894977

Kuan Peng wrote:
> Le 27/01/2026 à 22:45, Thomas 'PointedEars' Lahn a écrit :
>> Kuan Peng wrote:
>>>> What exactly do you mean by that?
>>
>> You have not answered my question.
>>
> I mean that the law
>>   ∇ × E' = -∂B'/∂t,                                       (2)
> is the same for all physicists.

Yes, but you confused the questions.  I asked about this:

>>>>> If at the beginning the current in A is constant,
>>>> What exactly do you mean by that?

> We understand it as that the curl of the electric field equals minus
> the rate of change of the magnetic field.

Yes.

> But each physicist attribute it different sense in his head and it is his 
> proper interpretation that makes that different physicist when predicting 
> the outcome of a same experiment using the same formula ∇ × E' = 
> -∂B'/∂t can give different result.

No.
> Le 27/01/2026 à 22:37, Thomas 'PointedEars' Lahn a écrit :

If you had only replied to one posting as you should have, you would
probably not have confused the questions.

>> I wonder how there is a current in (the secondary) coil B at all *before*
>> electromagnetic induction.  Usually there is not, i.e. the secondary coil
>> is NOT connected to a voltage source, but to some electric appliance:> 

> 1.	Before the current circulates in coil A, there is not a current in coil 
> B .
> 2.	After the current circulates in coil A, a current is induced in coil B.

In practice there is no "circulating current" as it is an "alternating"
current to maximize the induced current.

Also, in general you should not think of electricity as electrons (or worse,
positive charges) flowing through a conductor from one end of a circuit to
the other, like flowing water.  That is NOT how it works:

Veritasium: The Big Misconception About Electricity
<https://youtu.be/bHIhgxav9LY>

> 3.	Now, we have a current in coil A and a current in coil B.

So far we are (partially) in agreement, then.

> 4.	Because the current in coil A increases linearly, the magnetic field in 
> coil B increases linearly

This could only be said of the _average strength_ of the magnetic field (but
it would be wrong regardless, see below).  The magnetic *field* is a
*vector* *field*; it does not make sense to say that a field increases,
especially not a vector field.

> and its rate of change is -∂B'/∂t which is constant.

No.  You have to consider that any change of the magnetic field also induces
a current that flows opposite the current that produced the non-zero field
values -- Lenz's Law -- which I had indicated by putting primes in the
*second* induction equation.

> 5.	Because-∂B'/∂t is constant, ∇ × E' = -∂B'/∂t is constant. 
> [...]

/Ex falso quodlibet./

-- 
PointedEars

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

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#894982

Le 29/01/2026 à 04:19, Thomas 'PointedEars' Lahn a écrit :
>> Le 27/01/2026 à 22:37, Thomas 'PointedEars' Lahn a écrit :
>>> I wonder how there is a current in (the secondary) coil B at all *before*
>>> electromagnetic induction.  Usually there is not, i.e. the secondary coil
>>> is NOT connected to a voltage source, but to some electric appliance:> 
> 
>> 1.	Before the current circulates in coil A, there is not a current in coil 
>> B .
>> 2.	After the current circulates in coil A, a current is induced in coil B.
> 
> In practice there is no "circulating current" as it is an "alternating"
> current to maximize the induced current.
By " Before the current circulates in coil A "  , I mean " Before the 
current occurs in coil A " 

You think the current to be "alternating", I think it to be direct, 
although increasing current. This is an example of " each physicist 
attribute it different sense in his head ", we are not talking about the 
same experiment while talking about the same Faraday’s law. 
> 
> Also, in general you should not think of electricity as electrons (or worse,
> positive charges) flowing through a conductor from one end of a circuit to
> the other, like flowing water.  That is NOT how it works:
Is electron beam an electric current? Do we use Maxwell’s equations to 
describe its behavior? 
> 
> Veritasium: The Big Misconception About Electricity
> https://youtu.be/bHIhgxav9LY
I have viewed this video a while ago. I think he is not a specialist of 
electromagnetism.
Here too, we see the same video but have different interpretation. 

>> 4.	Because the current in coil A increases linearly, the magnetic field in 
>> coil B increases linearly
> 
> This could only be said of the _average strength_ of the magnetic field (but
> it would be wrong regardless, see below).  The magnetic *field* is a
> *vector* *field*; it does not make sense to say that a field increases,
> especially not a vector field.
The flux of magnetic field is a scalar and can increase. The induced 
voltage is proportional to the rate of increase of the flux. 

> 
>> and its rate of change is -∂B'/∂t which is constant.
> 
> No.  You have to consider that any change of the magnetic field also induces
> a current that flows opposite the current that produced the non-zero field
> values -- Lenz's Law -- which I had indicated by putting primes in the
> *second* induction equation.
The rate of change of the flux is constant, then the induced voltage is 
constant. 

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#894984

Kuan Peng <titang78@gmail.com> wrote or quoted:
>You think the current to be "alternating", I think it to be direct, 
>although increasing current.

  Maxwell's equations hold in any case.

>Is electron beam an electric current?

  Yes. The current is the charge per unit time passing a given point.
  Negative charges count as positive charges in the opposite direction.

>                                      Do we use Maxwell’s equations to 
>describe its behavior? 

  The behavior (acceleration) of each electron is given by
  Newton's second law, F=ma, where F = F_em + F_other, and the
  electromagnetic force F_em = Q[E + (v x B)] (when radiation
  reaction [the recoil due to the electron's own electromagnetic
  radiation] is negligible, which it usually is).

  

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#894996

Le 29/01/2026 à 23:08, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>   The behavior (acceleration) of each electron is given by
>   Newton's second law, F=ma, where F = F_em + F_other, and the
>   electromagnetic force F_em = Q[E + (v x B)] 
Is this correct? or 

Le 29/01/2026 à 23:13, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>I do not deem this to be an actual paradox, because we can
>   clearly see that here electromagnetism has priority over Newton.

This is correct 

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#894998

Kuan Peng <titang78@gmail.com> wrote or quoted:
>Le 29/01/2026 à 23:08, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>>The behavior (acceleration) of each electron is given by
>>Newton's second law, F=ma, where F = F_em + F_other, and the
>>electromagnetic force F_em = Q[E + (v x B)] 
>Is this correct? or 
>Le 29/01/2026 à 23:13, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>>I do not deem this to be an actual paradox, because we can
>>clearly see that here electromagnetism has priority over Newton.
>This is correct 

  Newton's /first/ and /second/ laws are /always/ correct. 
  Only the third law is not valid in electrodynamics.

  Feynman discusses this in Volume II, at the end of 26-2 
  "The fields of a point charge with a constant velocity".

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#894999

Le 30/01/2026 à 20:04, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
> Kuan Peng <titang78@gmail.com> wrote or quoted:
>>Le 29/01/2026 à 23:08, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>>>The behavior (acceleration) of each electron is given by
>>>Newton's second law, F=ma, where F = F_em + F_other, and the
>>>electromagnetic force F_em = Q[E + (v x B)] 
>>Is this correct? or 
>>Le 29/01/2026 à 23:13, ram@zedat.fu-berlin.de (Stefan Ram) a écrit :
>>>I do not deem this to be an actual paradox, because we can
>>>clearly see that here electromagnetism has priority over Newton.
>>This is correct 
> 
>   Newton's /first/ and /second/ laws are /always/ correct. 
>   Only the third law is not valid in electrodynamics.
> 
>   Feynman discusses this in Volume II, at the end of 26-2 
>   "The fields of a point charge with a constant velocity".

I do not agree, but OK.

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#894986

 Kuan Peng writes:
> You think the current to be "alternating", I think it to be direct,
> although increasing current.

It is a superposition of a DC component and a triangle wave.  We can
ignore the DC component in the steady state.
-- 
John Hasler 
john@sugarbit.com
Dancing Horse Hill
Elmwood, WI USA

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#894988

John Hasler wrote:
>  Kuan Peng writes:
>> You think the current to be "alternating", I think it to be direct,
>> although increasing current.
> 
> It is a superposition of a DC component and a triangle wave.  We can
> ignore the DC component in the steady state.

Nonsense.

-- 
PointedEars

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

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#894993

Le 30/01/2026 à 01:08, John Hasler a écrit :
>  Kuan Peng writes:
>> You think the current to be "alternating", I think it to be direct,
>> although increasing current.
> 
> It is a superposition of a DC component and a triangle wave.  We can
> ignore the DC component in the steady state.
We can. But the paradox subsists. 

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#894987

Kuan Peng wrote:
> Le 29/01/2026 à 04:19, Thomas 'PointedEars' Lahn a écrit :
>>> Le 27/01/2026 à 22:37, Thomas 'PointedEars' Lahn a écrit :
>>>> I wonder how there is a current in (the secondary) coil B at all *before*
>>>> electromagnetic induction.  Usually there is not, i.e. the secondary coil
>>>> is NOT connected to a voltage source, but to some electric appliance:> 
>>
>>> 1.	Before the current circulates in coil A, there is not a current in coil 
>>> B .
>>> 2.	After the current circulates in coil A, a current is induced in coil B.
>>
>> In practice there is no "circulating current" as it is an "alternating"
>> current to maximize the induced current.
> By " Before the current circulates in coil A "  , I mean " Before the 
> current occurs in coil A " 
> 
> You think the current to be "alternating",

It is alternating.

> I think it to be direct, 

Merely an academic possibility.  In real life, a transformer transforms high
voltage to low voltage or vice-versa because the current is an alternating
current.

>> Also, in general you should not think of electricity as electrons (or worse,
>> positive charges) flowing through a conductor from one end of a circuit to
>> the other, like flowing water.  That is NOT how it works:
> Is electron beam an electric current?

Yes.

> Do we use Maxwell’s equations to describe its behavior? 

Yes.

Your point being?
>> Veritasium: The Big Misconception About Electricity
>> https://youtu.be/bHIhgxav9LY
> I have viewed this video a while ago. I think he is not a specialist of 
> electromagnetism.

You are not in a position to make an informed judgement because evidently
you have never studied physics.  He has, and so have I; he is correct.

> Here too, we see the same video but have different interpretation. 

I am not interpreting, I *know* because I have studied it.  *You* are
interpreting because you do NOT know.  Big difference.

>>> 4.	Because the current in coil A increases linearly, the magnetic field in 
>>> coil B increases linearly
>>
>> This could only be said of the _average strength_ of the magnetic field (but
>> it would be wrong regardless, see below).  The magnetic *field* is a
>> *vector* *field*; it does not make sense to say that a field increases,
>> especially not a vector field.
> The flux of magnetic field is a scalar and can increase. 

Yes.

> The induced voltage

Voltages are not induced.

> is proportional to the rate of increase of the flux. 

Only because of the electric field that is induced by the change of the
magnetic (flux density) field.  We have been over this already.

>>> and its rate of change is -∂B'/∂t which is constant.
>>
>> No.  You have to consider that any change of the magnetic field also induces
>> a current that flows opposite the current that produced the non-zero field
>> values -- Lenz's Law -- which I had indicated by putting primes in the
>> *second* induction equation.
> The rate of change of the flux is constant, then the induced voltage is 
> constant. 

Read again what I wrote.  *facepalm*

-- 
PointedEars

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

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