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Does Biot-Savart law apply in changing electric field or it needs a modification just like Ampere's law, to become as true as Maxwell's 4th equation?

In the update to Ampere's circuital law, the effect of changing electric field on magnetism was taken into account to produce Maxwell's 4th equation.

Now, any change in electric flux is caused by a moving charge. I realise that in a continuos current flowing in a circuit, the change in electric field is cancelled out.

So, my question is, since Biot-Savart law considers magnetism produced by small current elements dl, does this not make it as correct as the Maxwell's 4th equation? If not, then how is the magnetic field at a specific point defined correctly caused by a single moving charge?

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Does Biot-Savart law apply in changing electric field or it needs a modification just like Ampere's law, to become as true as Maxwell's 4th equation?

It does need modification. But it will never be the same as Maxwell, because Maxwell allows boundary conditions to determine the solutions, so it can allow electromagnetic waves. If you want to think of it as the magnetic field due to the current, then you have to accept that the total magnetic field can be different. But even then, it is not correct because it requires steady current, if you want to have currents that change, consider Jefimenko's equations for the field due to the past current and the past time rate of change in current.

In the update to Ampere's circuital law, the effect of changing electric field on magnetism was taken into account to produce Maxwell's 4th equation.

That makes it sound like changing electric fields cause magnetic fields. That makes it sounds like you can just change the electric field somewhere however you want and do that to make a magnetic field. That is not true. The circulation of the magnetic field in a far away loop is related to the flux of time derivatives of the electric field inside the area spanned by the loop even if the loop is far far away from where the electric field is changing. The circulating magnetic field far away is most definitely not caused by the electric field in the center change right now.

One can argue just as easily that the imbalance between the current density and the curl of the magnetic field causes the electric field to change.

Such a view assume that the electric and magnetic fields are real and have values throughout space (which makes sense since the electromagnetic field carries energy and momentum). And that somehow something must determine how the fields change, and one option is that the electric field changes because of the imbalance between the current density and the curl of the magnetic field and similarly the magnetic field changes because of the curl of the magnetic field.

Such a view merely requires that the fields and current density have values at points and that the rate the fields change can depend on the spatial variation of the field right there.

Now, any change in electric flux is caused by a moving charge.

Absolutely not true. You can have an electromagnetic wave travel through empty space, the curl of the magnetic field making the electric field change and the curl of the electric field making the magnetic field change, each producing the wave at a slightly different time based on the current spatial variation of the wave.

I realise that in a continuos current flowing in a circuit, the change in electric field is cancelled out.

If you mean a steady current isn't associated with a different kind of magnetic field then you are right. When the current density has a time rate of change at a moment $t=t_0$ this causes a spherical shell to expand at the speed of light and this shell has different electromagnetic fields on it than normal, in fact these fields get weaker at a slower rate than other fields so from far away these are the fields that dominate unless they cancel themselves out. Other sources of electric fields are charge density (which can exist with a steady current for instance you can have a charged wire, or have a neutral wire with a charge imbalance on the edges to help the current flow around a bend) and the time rate of change of charge density.

So, my question is, since Biot-Savart law considers magnetism produced by small current elements dl, does this not make it as correct as the Maxwell's 4th equation?

It is more accurate than we'd think. The reality is that the magnetic field due to currents has a part based on what the current used to be and part based on how it used to be changing. And the Biot-Savart is based in what the current is now, but in a way where if the current is changing slowly the net effect of the current then and the way it was changing then is very close to what Buot-Savart says about the current now.

It like you knew where you friend was and what direction they were running. So you can base something off where the were and how they were running, but you could also base it off where you extrapolate them to be now, and this can be very accurate. The correct result is to base it on the current back then and the rate of change of the current back then.

If the current isn't changing then the current back then is the current now and there is not change in current so Biot-Savart is correct for finding that portion of the magnetic field due to matter (as opposed to that due to the fields themselves). If the current is changing then the field now and the field then are different plus there is a whole new source, the rate of change of the current. But the net effect is as if you used the extrapolated current and so can be quite precise.

If not, then how is the magnetic field at a specific point defined correctly caused by a single moving charge?

The magnetic field at a moment is rightly caused by what it was at a slightly earlier moment changed by a curl of the electric field. Only something like that can account for the fact that fields can go through empty space without being caused by any charge ever (according to the Maxwell equation).

But you can think of the total electromagnetic field as a part due to the charges and currents and their time rates and as a part due to itself.

In that case you can find the magnetic field die to charges and currents at a point to be based entirely on current and the rate of change of current. And that it isn't based on those things now bit based on what they were back in the past. Specifically you can consider each bit of current sending out information at the speed of light in all directions and you base your magnetic field on what information has arrived just now. Similarly when current has a time rate of change you can consider that information being sent out at the speed of light in all directions and you base your magnetic field here and now on what information has arrived here just now.

So current from farther away affected you based on what they were doing way back then.

And while changing currents affect electric fields, charge and time rates of charge do not affect magnetic fields, for instance a neutral wire that stays neutral doesn't affect the magnetic field. If it changes to be not neutral then for the charge to change there must be current and the magnetic field will only care about that current (and the rate it changes)

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The Biot-Savart law applies not only to static fields, but also to cases where the electric field can be expressed as gradient of potential. This is possible in cases where the currents change slowly enough (the frequency of oscillations is low).

A.P. French, J.R. Tessman, Displacement Currents and Magnetic Fields, Am. J. Phys. 31, 201 (1962), http://dx.doi.org/10.1119/1.1969359

The Biot-Savart formula is not equivalent to the Maxwell equation

$$ \nabla \times \mathbf B = \mu_0 \mathbf j + \mu_0\epsilon_0 \frac{\partial \mathbf E}{\partial t} $$

because the latter has more solutions and these are different from the field given by the former.

We believe Maxwell's equation is more general.

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