How Can I Derive the Biot-Savart Law?

[kipper]

Hi Folks,
I have a quick question about the Biot-Savart Law, I know what it is, but I don't know how to derive it. Could anyone give me either links or show me the derivation with origins. I have tried to google it but to no avail.
Thanks in advance for all your help in advance
Kipper

 

[Peeter]

I believe you want to start with the Ampere-Maxwell equation and consider a "constant" current through a current loop. "constant" will mean that you can drop the $\partial E/\partial t$ term.

 

[Bob S]

Look at Chapter V of Electromagnetism by Slater and Frank (McGraw Hill 1947). Copies of the 1969 edition are selling for $5.33 and up at Amazon.com.

 

[Peeter]

A quick overview of a derivation, starting with the steady state Ampere-Maxwell equation:

$$\nabla \times \vec{B} = \mu_0 \vec{J}$$

with $\vec{B} = \nabla \times \vec{A}$, and the gauge choice $\nabla \cdot \vec{A} = 0$, with some vector equation manipulation can show:

$$\nabla^2 \vec{A} = -\mu_0 \vec{J}$$

This has the Poisson solution

$$\vec{A} = \frac{\mu_0}{4\pi} \int \frac{\vec{J'}}{\lvert \vec{r} - \vec{r}'\rvert} dV'$$

For this steady state current case, the integral over all space here can be restricted to the current loop. Roughly, speaking with $J' dV' = I \vec{\hat{j}}' dl$, computation of the curl of $\vec{A}$ gives the Biot-Savart law.

$$\vec{B} = \frac{\mu_0 I}{4\pi} \int dl' \hat{j}' \times \frac{\vec{r} -\vec{r}'}{{\lvert \vec{r} - \vec{r}'\rvert}^3}$$

 

[kipper]

Thanks for all the replies so far, so am I right in thinking that the Biot-Savart law applies only when we assume that there is stationary current?
Thanks for the start on that proof, I need to look further into Poisson solutions in Electromagnetism to understand that step.
Thanks once again

 

[Peeter]

so am I right in thinking that the Biot-Savart law applies only when we assume that there is stationary current?

Yes.

ps. Re the Poisson solution. You may actually be familiar with this from electrostatics (subst, \rho, \phi, \epsilon_0, ...)

 

[Cyosis]

Biot-Savart's law is an empirical law, just like Coulomb's law and Newton's law of gravitation. It is from these laws that Ampere's law, Gauss' law and Gauss' law from gravitation are derived. So to derive the law you sort of need a more fundamental law, which is as of yet not known to mankind.