1. Feb 3, 2016

greswd

Ampere's Law can be derived from the Biot-Savart Law.

Faraday's Law is similar to Ampere's Law.

Is there a "Biot-Savart equivalent" of Faraday's Law?

I imagine it might look something like this: (not taking into account Coulomb's Law)
$$\frac{d\vec{E}}{dV}=\frac{-\left(\frac{∂\vec{B}}{∂t}\right)×\vec{1_r}}{4\pi r^2}$$

Biot-Savart Law from hyperphysics:

Last edited: Feb 3, 2016
2. Feb 3, 2016

The Biot-Savart's law type solution for E, comes from curl E=-dB/dt (Maxwell/Faraday equation in differential form) would need to be an integral that extended everywhere. With curl B=uoJ (for dE/dt=0), the solution for B in integral form is Biot-Savart's law (in integral form) which can be isolated to a single electrical charge, making Biot-Savart's law for this single charge. With an integral solution for curl E, the dB/dt can not be similarly isolated, so that a Biot-Savart type form would likely be impractical. Proceeding with a Stokes theorem approach and getting an integral of E*dl around a closed loop seems to be the only practical route for this one.

Last edited: Feb 3, 2016
3. Feb 3, 2016

greswd

That appears to be the case. But I'm wondering if there is anything in the literature about such a formula or something similar.

Last edited: Feb 3, 2016
4. Feb 3, 2016

Even in the textbooks, they seem to often overlook the Biot-Savart integral form as being a solution to the (Maxwell's) curl equation. One thing that can arise with such a solution, is that a homogeneous solution needs to be added to the(inhomogeneous) integral solution. e.g. for curl B, I don't think it normally needs an an extra homogeneous solution, but I have found when using it to solve the curl H equation, that a solution of curl H=0 needs to be included to get a complete answer. (The integral H Biot-Savart type solution contains no magnetic pole contribution to H, and the curl H=0 solution contains this magnetic pole contribution.)

5. Feb 3, 2016

greswd

I see. So I'm wondering if there is anything in the literature about such a formula or something similar.

It also explains why an induced electric field can exist in a region where there is no magnetic field.

Last edited: Feb 3, 2016
6. Feb 3, 2016

Such a formula is useful, as you said, to show the non-local contributions of the dB/dt to the Faraday E, but I think it would be impractical to use it in a computation to calculate the Faraday E. e.g. if you had a sinusoidal current in a solenoid creating a changing magnetic field B, instead of using an integral E*dl=-d(magnetic flux)/dt, and using the symmetry to get integral=E*2*pi*r, try computing the E from the dB/dt everywhere in a Biot-Savart type integral solution for E. I think you might find the integrals to be unmanageable.

7. Feb 3, 2016

greswd

But I'm wondering if there is anything in the literature about such a formula or something similar?

8. Feb 3, 2016

In general, I think they have overlooked the Biot-Savart integral solution to the curl equation in the textbooks. The more advanced textbooks do show the integral solution to the div E equation (an inverse square integral form, basically Coulomb's law), but they don't normally present the curl equations in such a fashion and they really should. I only recently ( a couple years ago), recognized the integral solution to the curl equation, (the Biot-Savart form), upon an extensive review of the E&M subject. One other important concept that is often overlooked in the E&M textbooks is the magnetic surface currents. Griffiths book presents them reasonably well, but older textbooks often overlooked them entirely. (e.g. textbooks which emphasized the magnetic pole method.)

9. Feb 3, 2016

greswd

I believe the equation I presented is not related to Coulomb's law.

Are you saying that you have not encountered such a formula or something similar in the literature?

10. Feb 3, 2016

11. Feb 3, 2016

In the div E Maxwell differential equation, they do present the integral solution (of a Coulomb's law form) in many E&M textbooks. They do not normally do this for the curl B or curl E equation, even though I think they would do well to present it this way. (Incidentally, also not presented in the textbooks is an obstacle that is encountered with the H integral solution for the div H solution. Starting with B=H+4*pi*M, and take divergence of both sides, div H=-4*pi*div M Now -div M=magnetic charge density, so that integral solution for H (analogous to an electrostatic E-field) is in the Coulomb's law form for magnetic charges. This H solution turns out to be incomplete, because this time, the solution for H missed the H contribution from currents in conductors. That part, it turns out, is picked up by the homogeneous solution, div H=0 which must be included to have a complete answer for H. This generally is also not found in the E&M textbooks.) Now, for some feedback on your observation to use a Biot-Savart type formula in solving for E for Maxwell's/Faraday curl E=-dB/dt equation. (M.K.S. units) I do think it is a very keen observation on your part, especially if you are only at the undergraduate level=you are off to a very good start at getting a good handle on the subject.

12. Feb 3, 2016

greswd

Thanks. I wrote that equation because I wanted to find out about whether we can derive the electric field function from a changing magnetic field.

Ok, I suppose the answer to my question is 'yes' then.

13. Feb 3, 2016

Yes, the E field can be obtained from the Biot-Savart type equation for a changing magnetic field (curl E=-dB/dt is an inhomogeneous differential equation). In a system in which static electric charges are also present, you will still need to add a solution to the homogeneous equation, a solution of E for curl E=0 to pick up the homogeneous solution for the part of the E-field that comes from static electric charges in order to have the complete and correct expression for the E-field. (This homogeneous solution is in fact the Coulomb's law expression for the static electric charges.)

14. Feb 3, 2016

15. Feb 4, 2016

Please see my reply #10 and also my reply #13. A couple additional comments-Even in computing the the electric field E from electrical charges using the Coulomb's law type integral solution for the div E equation, the solution for E is not complete if there are any changing magnetic fields. The Faraday E needs to be included and that enters in as the solution of the homogeneous equation div E=0, which is precisely the Biot-Savart type solution for E that you presented in your initial post at the top.

16. Feb 5, 2016

Staff: Mentor

Start with the volume-integral version of the Biot-Savart law for a current density $\vec J$: $$\vec B = \frac{\mu_0}{4\pi} \int {\frac {\vec J \times \vec r^\prime}{|\vec r^\prime|^2} dV}$$ Recall that we can write the complete Ampere-Maxwell law using Maxwell's "displacement current density" $\vec J_d$: $$\vec \nabla \times \vec B = \mu_0 \vec J + \mu_0 \epsilon_0 \frac {\partial \vec E}{\partial t} = \mu_0 (\vec J + \vec J_d)$$ where $$\vec J_d = \epsilon_0 \frac {\partial \vec E}{\partial t}$$ This suggests that we can get a "complete" version of the Biot-Savart law by replacing $\vec J$ with $\vec J + \vec J_d$.

Now, what you want is an "electric Biot-Savart law" corresponding to Faraday's law. If magnetic charge existed, Faraday's law would have a "magnetic current" $\vec J_m$ on the right-hand side, along with the $\frac {\partial \vec B}{\partial t}$ term which would correspond to a "magnetic displacement current" $\vec J_{md}$ analogous to the displacement current mentioned above. The corresponding "electric Biot-Savart law" would then look something like $$\vec E = K \int {\frac {(\vec J_m + \vec J_{md}) \times \vec r^\prime}{|\vec r^\prime|^2} dV}$$ Of course, magnetic charge doesn't exist, so $\vec J_m = 0$, leaving only $\vec J_{md}$. I leave it to you to fill in the details and figure out the appropriate constant(s).

Last edited: Feb 5, 2016
17. Feb 15, 2016

greswd

@ jtbell is it the same as the one I first posted? Seems to be the case.

I would like to know if there is anything in the literature on an electric Biot-Savart law.

18. Feb 15, 2016

greswd

ok, but I'm sorry, I can't find any reference to a "Faraday" Biot-Savart law in your write up.

19. Feb 15, 2016