Is the Biot-Savart Law reversible?

[Dale]

Only its position varies, not its absolute magnitude, which is measured relative to its origin and not relative to some arbitrary point or another charge.

That is still a nonzero ∂B/∂t everywhere. Biot Savart is derived from Maxwell's equations under the assumption that ∂B/∂t=0 (magnetostatic).

You cannot just blindly apply formulas. You need to know what the variables mean and what the underlying assumptions are.

 

[carrz]

That is still a nonzero ∂B/∂t everywhere. Biot Savart is derived from Maxwell's equations under the assumption that ∂B/∂t=0 (magnetostatic).

Biot Savart is for magnetic fields what Coulomb's law is for electric fields. Biot-Savart law applies wherever Coulomb's law applies. Biot Savart law was formulated 10 years before Maxwell was born.

What is the deviation of Biot Savart or Coulomb's law for velocities of 2/3 the speed of light?

You cannot just blindly apply formulas. You need to know what the variables mean and what the underlying assumptions are.

I thought it's well known Biot-Savart law is commonly used for many practical situations relating to free electrons, electron beams, and electric currents just the same. It's used for cathode ray tubes for example, for trajectories in bubble chambers, for "bunching" in electron beams, for "z-pinch" of electron plasma, and it's used as the basis of operation for many, if not all, electronic instruments and components because of its relation to ampere unit. I really don't see what practical situation would require Lienard Wiechert potentials, can you give me some examples where are those equations used in practice?

 

[vanhees71]

Of course, also Coulomb's Law is only valid for static charge distributions as is the Biot-Savart Law only valid for stationary current distributions. Again: For a moving point charge, neither the charge nor the current distribution are time-independent, and thus neither Coulomb's nor Biot-Savart's Law are applicable in this situation, no matter how fast the particle is moving.

Of course, before Maxwell's discovery of the full electromagnetic field equations, the knowledge about electromagnetism was incomplete. You cannot argue with an invalid predecessor theory (Weber's and Ampere's action-at-a-distance models in that case) as it is known to be inapplicable for the given situation.

 

[Jano L.]

Biot-Savart law applies exactly to "not-stationary" charges, that's why there is 'velocity' in the equation, the magnitude of magnetic field depends on it.

Of course, also Coulomb's Law is only valid for static charge distributions as is the Biot-Savart Law only valid for stationary current distributions. Again: For a moving point charge, neither the charge nor the current distribution are time-independent, and thus neither Coulomb's nor Biot-Savart's Law are applicable in this situation, no matter how fast the particle is moving.

The truth is somewhere in between. If we assume that the Maxwell equations are always obeyed, the Biot-Savart law
$$
\mathbf B(\mathbf x) = \frac{\mu_0}{4\pi} \int \frac{\mathbf j(\mathbf x')\times |\mathbf x- \mathbf x'|}{|\mathbf x- \mathbf x'|^3}\,d^3\mathbf x'
$$
is valid for stationary currents as well as for non-stationary currents if the electric field is given by gradient of potential.

One moving particle obeys these conditions provided it moves much slower than $c$. For fast moving or accelerating particle, the general Heaviside-Feynman formulae should be used.

See the end of the page, formula 6.59
http://www.physics.buffalo.edu/phy514/w02/index.html

 

[carrz]

The truth is somewhere in between.

In between, or is it rather close to speed of light? For example, what is the deviation of Coulomb's law for velocities of 3/4 speed of light? I can't google anything about it except a few obscure papers which rather suggest deviation was not experimentally confirmed.

is valid for stationary currents as well as for non-stationary currents if the electric field is given by gradient of potential.

That's right. For plotting tracks of point charges that would be instantaneous velocity integrated over acceleration points along the path.

 

[carrz]

Of course, also Coulomb's Law is only valid for static charge distributions as is the Biot-Savart Law only valid for stationary current distributions. Again: For a moving point charge, neither the charge nor the current distribution are time-independent, and thus neither Coulomb's nor Biot-Savart's Law are applicable in this situation, no matter how fast the particle is moving.

Of course, before Maxwell's discovery of the full electromagnetic field equations, the knowledge about electromagnetism was incomplete. You cannot argue with an invalid predecessor theory (Weber's and Ampere's action-at-a-distance models in that case) as it is known to be inapplicable for the given situation.

Can you give me a practical example where Biot-Savart and Coulomb's law do not apply and some other equations are used instead?

 

[Matterwave]

Can you give me a practical example where Biot-Savart and Coulomb's law do not apply and some other equations are used instead?

Anywhere there's not electromagneto-statics Maxwell's equations are used instead.

A single accelerating charge, for example, will not produce electric and magnetic fields according to the Coulomb and Bio Savart laws.

 

[carrz]

Anywhere there's not electromagneto-statics Maxwell's equations are used instead.

Sounds vague to me. Would you say deflecting electrons in a cathode ray tube or z-pinch effect of electron plasma, like in lightning bolts, is electromagneto-statics?

Biot-Savart law is used for calculating magnetic forces acting on electric charges in motion, attraction and repulsion, together with Lorentz force equation. Nothing else. But whether those charges have constant velocity or not is irrelevant, it's only a matter of differential kinematic equations.

A single accelerating charge, for example, will not produce electric and magnetic fields according to the Coulomb and Bio Savart laws.

Are you saying two electrons 1 millimeter apart will not repel with the same Coulomb force when they are accelerating as they would if they were moving at constant velocity? What theory are you referring to?

 

[Matterwave]

Why is it vague? Any time you have time varying electric and/or magnetic fields, then you need to use more general equations. This is because a time varying electric field will produce a magnetic field (Modified Ampere's law) and a time varying magnetic field will produce an electric field (Faraday's law).

Two electrons 1 millimeter apart will not repel with the same force if they are accelerating, as 2 stationary electrons 1 millimeter apart. Both electrons will see a magnetic in their respective frames. In addition, the electric field due to both particles will be modified by acceleration terms. The Lienard Wiechert potentials encapsulate this aspect of E&M perfectly as explicit terms with $\dot{v}$ appear in both the electric and magnetic field expressions.

 

[carrz]

Why is it vague? Any time you have time varying electric and/or magnetic fields, then you need to use more general equations. This is because a time varying electric field will produce a magnetic field (Modified Ampere's law) and a time varying magnetic field will produce an electric field (Faraday's law).

Electrons in a cathode ray tube, are their electric and magnetic fields time varying?

Two electrons 1 millimeter apart will not repel with the same force if they are accelerating, as 2 stationary electrons 1 millimeter apart.

Can you point a reference that confirms such deviation has been experimentally verified?

 

[vanhees71]

The quasi-stationary approximation is valid in regions close to the sources, i.e., at distances smaller than the typical wavelength of an electromagnetic field. In this "near-field zone" you can neglect retardation. That's why the quasitationary approximation works for usual household AC.

Again, the charge and current density of a single charge is given by
\rho(t,\vec{x})=q \delta^{(3)}[\vec{x}-\vec{y}(t)], \quad \vec{j}(t,\vec{x})=q \dot{\vec{y}}(t) \delta^{(3)}[\vec{x}-\vec{y}(t)],
where \vec{y}(t) is the trajectory of the particle in a fixed reference frame. You clearly see that even for a uniformly moving particle these are not stationary, and you have to use the retarded expressions to get the correct field. In that case you can also use the trick with the Lorentz boost, I've demonstrated some postings before.

 

[carrz]

The quasi-stationary approximation is valid in regions close to the sources, i.e., at distances smaller than the typical wavelength of an electromagnetic field. In this "near-field zone" you can neglect retardation. That's why the quasitationary approximation works for usual household AC.

So there are two different scenarios where standard Biot-Savart and Coulomb's law deviate:

a. charges have non-uniform velocity

b. charges are moving at high velocity


Has this been experimentally confirmed, and can you point some reference about it?

 

[vanhees71]

Maxwell electrodynamics and its quantized version, QED, can be considered as the best tested theory ever in the sense that some fundamental properties like the (anomalous) magnetic moment of the electron is confirmed to agree with the value predicted by theory of 13 significan decimal places. For a first orientation on this, see

http://en.wikipedia.org/wiki/Precision_tests_of_QED

Concerning the classical limit, the functioning of high-energy particle accelerators like the LHC at CERN also shows that Maxwell electrodynamics and relativistic (!) particle dynamics works with high precision.

Relativistic effects are not always related with large speeds. E.g., the homopolar generator can only be understood when the relativistic structure of the theory is used, and there no high speeds (compared to the speed of light in vacuo) is involved. Have a look here

http://en.wikipedia.org/wiki/Faraday_paradox

Last but not least, any device using e.m. waves to transmit signals (radios, cell phones, etc.) prove the correctness of Maxwell's prediction of electromagnetic waves.

 

[carrz]

Maxwell electrodynamics and its quantized version, QED, can be considered as the best tested theory ever in the sense that some fundamental properties like the (anomalous) magnetic moment of the electron is confirmed to agree with the value predicted by theory of 13 significan decimal places. For a first orientation on this, see

http://en.wikipedia.org/wiki/Precision_tests_of_QED

Concerning the classical limit, the functioning of high-energy particle accelerators like the LHC at CERN also shows that Maxwell electrodynamics and relativistic (!) particle dynamics works with high precision.

Relativistic effects are not always related with large speeds. E.g., the homopolar generator can only be understood when the relativistic structure of the theory is used, and there no high speeds (compared to the speed of light in vacuo) is involved. Have a look here

http://en.wikipedia.org/wiki/Faraday_paradox

Last but not least, any device using e.m. waves to transmit signals (radios, cell phones, etc.) prove the correctness of Maxwell's prediction of electromagnetic waves.

Biot-Savart and Lorentz force have nothing to do with radiation or em waves, just force. They do not compare with Maxwell equations, not all of them anyway. I'm not convinced experimental verifications of QED actually prove Lorenz or Coulomb force will vary depending on velocity. The only thing I found related to that actually suggests otherwise, that is experimental checks on photon mass are measured relative to Coulomb's law and depend on its constancy.

 

[Matterwave]

Biot-Savart and Lorentz force have nothing to do with radiation or em waves, just force. They do not compare with Maxwell equations, not all of them anyway. I'm not convinced experimental verifications of QED actually prove Lorenz or Coulomb force will vary depending on velocity. The only thing I found related to that actually suggests otherwise, that is experimental checks on photon mass are measured relative to Coulomb's law and depend on its constancy.

The Lorentz force is correct, even relativistically and even for accelerating charges. This is only a force law. The Coulomb's law is valid for only electro-statics because it depends on a static electric field to derive the force.

 

[Dale]

Closed pending moderation.