Is the Biot-Savart Law reversible?

[Dale]

It's not true. See Heaviside's electrodynamics.

It is true. Heaviside's electrodynamics includes more than the Biot Savart law.

Whether it is a single moving charge or a current in a wire, it will create a magnetic field which will move a stationary charge.

A single moving charge and a current in a wire are substantially different situations. A single moving charge has a time varying E-field and a time varying B-field and cannot be described by the Biot Savart law (a stationary charge will experience a net force due to the moving charge). A steady current in a wire has a static B-field but no E-field and can be described by the Biot Savart law (a stationary charge will not experience a net force due to the current).

 

[carrz]

No, it does not! A charge in motion creates a time-varying magnetic field given by the Lienard Wiechert potentials.

According to Biot-Savart law magnitude of magnetic field varies only with velocity, so if velocity is constant magnetic field stays constant. Why are you bringing in relativity when the question is explicitly about Biot-Savart law?


Can you explain how magnetic field varies when a charge is moving with constant velocity? Is it the magnitude that grows stronger and weaker, or something else, and what is this variation relative to?

 

[carrz]

A single moving charge and a current in a wire are substantially different situations. A single moving charge has a time varying E-field and a time varying B-field and cannot be described by the Biot Savart law (a stationary charge will experience a net force due to the moving charge). A steady current in a wire has a static B-field but no E-field and can be described by the Biot Savart law (a stationary charge will not experience a net force due to the current).

The ampere unit is defined by Biot-Savart law and Lorentz force, and their electric current equations are just integrals of their point charge equations, so it must be the same situation or the equations would not work and our unit of ampere would be wrong.

http://en.wikipedia.org/wiki/Ampère's_force_law

That is simply integral of Biot-Savart law and Lorentz force for a single moving charge:

These two equations apply to free electrons, electron beams, and electric wires just the same.

 

[Dale]

According to Biot-Savart law magnitude of magnetic field varies only with velocity, so if velocity is constant magnetic field stays constant. Why are you bringing in relativity when the question is explicitly about Biot-Savart law?

Huh? I am not bringing in relativity. The Lienard Wiechert potential applies for all speeds, and the deviations from Biot Savart and Coulomb's law are significant even for non relativistic speeds. It was even developed prior to relativity.

The B field varies because the distance to the moving charge varies, and the E field varies for the same reason. Furthermore, because of the temporal variation the magnitude and direction of the field differs from that predicted by Biot Savart.

Can you explain how magnetic field varies when a charge is moving with constant velocity? Is it the magnitude that grows stronger and weaker, or something else, and what is this variation relative to?

When the charge is far away there is little field. As the charge approaches and passes the field increases, changes direction, and decreases.

 

[vanhees71]

I brought in relativity for the special case of a uniformly moving charge, because it's much simpler to calculate the electromagnetic field with help of a Lorentz boost. Let's do that. In the lab frame (unprimed coordinates (x^{\mu})=(ct,\vec{x})) the particle moves with a constant velocity \vec{v}=v \vec{e}_x. We want to evaluate the field in this reference frame.

To that end we look at the situation from the reference frame, where the particle is at rest (primed coordinates (x'^{\mu}=(c t',\vec{x}')) at the origin. There the electromagnetic field is simply an electrostatic Coulomb field, which reads (in Heaviside-Lorentz units)
\vec{E}'(t',\vec{x}')=\frac{q}{4 \pi} \frac{\vec{x}'}{r'^3}, \quad \vec{B}'(t',\vec{x}')=0.
Now we use the transformation of the space-time coordinates and the em. field under the Lorentz boost
c t'=\gamma(c t-v x/c), \quad x'=\gamma(x- v t), \quad y'=y, \quad z'=z.
E_x(t,\vec{x})=E_x'(t',\vec{x}')=\frac{q}{4 \pi} \frac{x'}{r'^3}=\frac{q}{4 \pi} \gamma \frac{x-v t}{[\gamma^2(x-v t)^2+y^2 +z^2]^{3/2}},
E_y(t,\vec{x})=\gamma E_y'(t',\vec{x}')=\frac{q}{4 \pi} \gamma \frac{y}{[\gamma^2(x-v t)^2+y^2 +z^2]^{3/2}},
E_z(t,\vec{x})=\gamma E_z'(t',\vec{x}')=\frac{q}{4 \pi} \gamma \frac{z}{[\gamma^2(x-v t)^2+y^2 +z^2]^{3/2}}
or in three-vector notation
\vec{E}(t,\vec{x})=\frac{q}{4 \pi} \frac{\gamma}{[\gamma^2(x-v t)^2+y^2 +z^2]^{3/2}} \begin{pmatrix}<br /> x-v t \\ y \\ z \end{pmatrix}.<br />
The magnetic field is given by
\vec{B}=\vec{\beta} \times \vec{E}=\frac{q}{4 \pi} \frac{\gamma}{[\gamma^2(x-v t)^2+y^2 +z^2]^{3/2}} \begin{pmatrix}<br /> 0 \\ -z \\ y \end{pmatrix}.<br />
Of course, you get the same result by using the four-current
\rho(t,\vec{x})=q \delta(x-v t) \delta(y) \delta(z), \quad \vec{j}=q \vec{v} \delta(x-v t) \delta(y) \delta(z)
in the retarded (Lienard-Wiechert) potentials and taking the according derivatives
\vec{E}=-\frac{1}{c} \partial_t \vec{A}-\vec{\nabla} A^0, \quad \vec{B}=\vec{\nabla} \times \vec{A}<br /> or directly the retarded Jefimenko integrals for the field.<br /> <br /> You <b>must not</b> use the naive Biot-Savart Law for the magnetic field, because the current density is <b>not stationary</b>, because it&#039;s obviously time dependent!

 

[carrz]

You must not use the naive Biot-Savart Law for the magnetic field, because the current density is not stationary, because it's obviously time dependent!

Biot-Savart law applies exactly to "not-stationary" charges. We sure use it to define the ampere unit, which is calibration reference for many if not most of our electronic instruments and components. We are already using it, a lot. Do you believe there is even any difference when velocities are below the speed of light?

 

[carrz]

Huh? I am not bringing in relativity. The Lienard Wiechert potential applies for all speeds, and the deviations from Biot Savart and Coulomb's law are significant even for non relativistic speeds. It was even developed prior to relativity.

The question is explicitly about Biot-Savart law. If it was an exam I don't think your answer would earn any points. But since we are already talking about it, what is the deviation of Biot Savart or Coulomb's law for, say velocities of 2/3 the speed of light?

The B field varies because the distance to the moving charge varies, and the E field varies for the same reason. Furthermore, because of the temporal variation the magnitude and direction of the field differs from that predicted by Biot Savart.

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. If you want to talk about what some other charge experiences in the field of the first charge, then of course "B field varies" relative to that second charge, it varies inverse-proportionally to the square of the distance. That's exactly what Biot-Savart laws say as well, the only difference is semantics.

 

[vanhees71]

Biot-Savart law applies exactly to "not-stationary" charges. We sure use it to define the ampere unit, which is calibration reference for many if not most of our electronic instruments and components. We are already using it, a lot. Do you believe there is even any difference when velocities are below the speed of light?

The Biot-Savart Law is valid only for stationary fields. Electromagnetism is a local relativistic classical field theory, and there is no value in not treating it as such. A lot of apparent difficulties, particularly with Faraday's Law simply vanish when doing so. E.g., the homopolar generator becomes very simple to understand when treated relativistically.

The Biot-Savart Law is a special case of the Jefimenko equations for stationary charge-current distributions. That's it. The definition of the Ampere refers to a static situation. So there is no contradiction here.

 

[carrz]

The Biot-Savart Law is valid only for stationary fields. Electromagnetism is a local relativistic classical field theory, and there is no value in not treating it as such. A lot of apparent difficulties, particularly with Faraday's Law simply vanish when doing so. E.g., the homopolar generator becomes very simple to understand when treated relativistically.

The Biot-Savart Law is a special case of the Jefimenko equations for stationary charge-current distributions. That's it. The definition of the Ampere refers to a static situation. So there is no contradiction here.

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. Electrons are not static when two parallel wires repel or attract, they move quite a bit, and the faster they go, the greater is magnetic field they cause. Biot-Savart law and Lorenz force equations for point charges apply to free electrons, electron beams, and electric currents just the same. How would you like me to prove that to you?

 

[Dale]

Biot-Savart law applies exactly to "not-stationary" charges. We sure use it to define the ampere unit, which is calibration reference for many if not most of our electronic instruments and components. We are already using it, a lot.

When the Biot Savart law is used in this context the current density is constant wrt time, so it applies. I am not saying that it is never valid but rather that it only applies to magnetostatic situations, which a moving point charge is not but a wire with constant current is.

 

[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]

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