[hamdani yusuf]

In my example, Q is constant over time.

and it is moving

A 1 Coulomb charged particle moves at 1 m/s speed. What's the current?
In cgs system, the same experiment would read:
A 1 Coulomb charged particle moves at 100 cm/s speed. What's the current?

[Bored chemist]

A 1 Coulomb charged particle moves at 1 m/s speed. What's the current?

It depends
Imagine I put that coulomb into a 1 metre cube box. At 1 m/s the whole coulomb goes past me in 1 second and that's a current of 1 amp.
Now imaging I put the same charge in a box 10 metres long.
It now takes 10 seconds to go past me.
So that's 1 C in 10 S or 0.1 C/S so that's 0.1 amps.

You really need to study science a bit more in order to avoid asking meaningless question.

[hamdani yusuf]

A 1 Coulomb charged particle moves at 1 m/s speed. What's the current?

It depends
Imagine I put that coulomb into a 1 metre cube box. At 1 m/s the whole coulomb goes past me in 1 second and that's a current of 1 amp.
Now imaging I put the same charge in a box 10 metres long.
It now takes 10 seconds to go past me.
So that's 1 C in 10 S or 0.1 C/S so that's 0.1 amps.

You really need to study science a bit more in order to avoid asking meaningless question.

It's meaningless to you because you haven't understood the problem yet. It shows that Maxwell's equations are not adequate to describe electrodynamics systems.

Let's distribute the electric charge into a thin metal disc with 10 m diameter and 0.1 mm thick. The disc moves axially at 1 m/s. What's the electric current?

Here's another example.
Electrons move in a CRT at approximately 0.1 c. What's the current generated by each electron?  What's the expected B field at a point 1 mm from the trajectory of the electron?

[Bored chemist]

It's meaningless to you because you haven't understood the problem yet.

Here's is the problem you set.

A 1 Coulomb charged particle moves at 1 m/s speed. What's the current?

What part of it do you think I didn't understand.
More importantly, what do you think the answer is?

[Bored chemist]

What's the current generated by each electron? 

Can I ask you to do something that will help a lot.
Before you post anything here, can you just check that the units of your question make sense.

Your question is as pointless as if you asked "How fast is a mile?"

[alancalverd]

Current is dQ/dt, however you choose to distribute Q.

[Bored chemist]

Current is dQ/dt,

Good point.

What's the current generated by each electron?

Charge at start of experiment about 10^-19 Coulombs
Charge at end of experiment about 10^-19 Coulombs
Change in charge
Zero

Rate of change of charge 
Zero

Current
Zero

Usefulness of question
Pretty close to zero.

[alancalverd]

But at any point in the flight path of the electron, 1ε of charge passed at 0.1c, so there was a current flowing from cathode to anode. Given a sufficiently sensitive compass needle, you could have detected the passage of a single electron by the induced magnetic pulse. The principle is used in the "ballistic galvanometers" that used to grace physics labs, and the integrating "mAs" meters on old x-ray machines.

Usefulness? not a lot, but the inverse phenomenon was used to deflect the electron beam in CRT televisions, radar displays and scanning electron microscopes, though magnetic drive is generally too slow for use in oscilloscopes.

[Bored chemist]

Ballistic galvanometers are calibrated in units of charge rather than current.

e.g "the galvanometer sensitivity is 2 divisions per microcoulomb."

From
https://gulpmatrix.com/the-working-principle-and-uses-of-a-ballistic-galvanometer/#gsc.tab=0


And one thing that makes then useful is that sometimes, the current vs time plot is complicated, but you only need to know the total charge (e.g testing magnets).

So, yes, you can in principle measure a charge of a single electron but, like the mA. Sec meter, it's measuring charge not current.

You can actually measure currents so small that, on average, there's less than 1 electron per second flowing through a circuit.
https://download.tek.com/document/2648%20Counting%20Electrons1.pdf
But the "per second" bit is important.

And... it's a bit fiddly.

[Eternal Student (OP)]

Hi.

   In regards to @hamdani yusuf  asking questions about a magnetic field due to a single moving charge.
    You were deceived a little and that's all.  @alancalverd provided a simplified calculation of the B field as if the moving charge was an ordinary current in a wire (I think,  not sure, it was a lot of posts ago).
   For a more complete answer you would need to consider all of Maxwell's laws and then the Biot-Savart law will emerge.   Note that there are TWO different things that are often called the Biot-Savart law.  We want the Biot-Savart law for a single moving charge and not for a steady current flowing in a wire.
   See  https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law#Point_charge_at_constant_velocity.
At low velocities the non-relativistic version is often used:

   

[Equation 1]

where vectors and scalars appear as usual:     is the velocity of the charged particle.  r is the scalar magnitude of r which is almost the position vector of the moving charge.  r is the vector to the moving charged particle from the point where you are trying to determine the B field.  So if you are determining B at position  r'  then  r = R - r'  with R = position of the charged particle (from the origin).      is the unit vector in the direction orthogonal to both v and  r (it's positive in the direction  v x r ).

    As you can see in [Equation 1]  the current,  I,  does not appear,  only the charge and the velocity of that charge.

- - - - - - - -
     As for the multiple questions about the current due to some moving charges with constant velocity.   For a genuine point charge (with charge Q),  the current becomes a Dirac delta function but Maxwell's equations do hold quite well using these functions.  Specifically I = current at position x  and time t   is defined as:

I(x  , t) =                  0           if the charge is never at position x.
                                Q. δ(t-t₀)       if the moving charge is at position x at time t₀.

   More usually we'd be interested in a current density vector, J, rather than just a current.   You hardly need to make use of such a peculiar definition anyway.   When applying Ampere's law (to the situation with just the one moving point charge), just choose a surface which does not have the moving charge on it at the time t.   Then J = 0 everywhere on that surface and the contribution to the magnetic field is then entirely due to the changing E field over that surface.    If you're interested, this is almost precisely the opposite conditions that apply for a steady current flowing in a wire and that version of the Biot-Savart law.   For a steady current in a wire, the E field would be unchanging and only the flow of charges matter and contribute to the magnetic field.  In the situation with just one charge, the E field is definitely changing (as the source moves) but the current density vector is 0 almost everywhere.    The fact that the Biot-Savart law for a steady current in a wire does look like the Biot-Savart law for a single moving charge is almost a happy co-incidence, indeed you can quickly prove one from the other (if you assume the magnetic field from a current in a wire is just the sum of the magnetic fields due to individual charges moving in that wire).   It's not a pure co-incidience, of course, but discussing that probably needs a different thread.

Best Wishes.

[hamdani yusuf]

See  https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law#Point_charge_at_constant_velocity.

The equations in the link are:


For electron in CRT, the equations below don't apply.

and


The most obvious limitation of Maxwell's equations is lacking of explanation for permittivity and permeability of various media. How can different distribution of electric charges change the permittivity and permeability of a point in space?

[Bored chemist]

The most obvious limitation of Maxwell's equations is lacking of explanation for permittivity and permeability of various media.

That's not their job.

[Eternal Student (OP)]

Hi.

For electron in CRT, the equations below don't apply.

   Why not?   
    They can be quick, up to 1/10 c according to one piece of text.   Have you tried the relativistic versions?
Also usually a CRT device like an oscilloscope doesn't try to determine or measure the magnetic and electric field generated by the ray. 

Best Wishes.

[paul cotter]

Permittivity and permeability are properties of various materials. Maxwell's equations deal with the behaviour of fields, not with material properties, though these properties do influence the maths.

[hamdani yusuf]

The most obvious limitation of Maxwell's equations is lacking of explanation for permittivity and permeability of various media.

That's not their job.

That's why they don't work well at microscopic scale.

[hamdani yusuf]

Hi.

For electron in CRT, the equations below don't apply.

   Why not?   
    They can be quick, up to 1/10 c according to one piece of text.   Have you tried the relativistic versions?
Also usually a CRT device like an oscilloscope doesn't try to determine or measure the magnetic and electric field generated by the ray. 

Best Wishes.

For high speed charged particles, retardation needs to be accounted. That's why the article mentioned Jefimenko.

[hamdani yusuf]

Permittivity and permeability are properties of various materials. Maxwell's equations deal with the behaviour of fields, not with material properties, though these properties do influence the maths.

Materials are made of electrically charged particles. Permeability and permittivity emerge from their distribution in space.

[Eternal Student (OP)]

Hi.

For high speed charged particles, retardation needs to be accounted. That's why the article mentioned Jefimenko.

   Yes, Jefimenko's equations are better for very high speed charged particles.  

    However, you weren't originally asking questions about high speed particles   (posts #148 through to #174 started with particles moving at ~ 1m/s and even the upgrade to Cathode rays had velocities ~ 0.1c).  I'm not sure what the problem was or where you were going next.  I had guessed it was about charge being infinitely divisible but perhaps it wasn't.   Anyway, that's perfectly fine  (a forum should have the discussion move and change direction for all sorts of reasons - otherwise what's the point of having a forum?)

Best Wishes.

[paul cotter]

I understand what Hamdani is alluding to and it is something I have often thought about-permeability and permittivity are macroscopic properties derived fundamentally from the presence of charges in said material. Do they have a meaning at the atomic level? Have to run now, working today, mv switchroom in a stinking meat plant-yuck.

[alancalverd]

That's why they don't work well at microscopic scale.

citation needed
AFAIK any moving charge creates a magnetic field, and any changing magnetic field can induce a current in a conductor. I've only worked with atomic nuclei (in MRI systems) but BC may well have played with electrons (chemists like ESR measurements). How microscopic did you have in mind? 

Back in the mists of my youth I was involved in the control and measurement of 400 keV electron beams. The relativistic effects were clearly demonstrable but I can't remember what fraction of c - we set it up eventually as an undergraduate teaching experiment.