# Deriving the Biot-Savart Law: Help Needed

• kipper
In summary: As for the derivation, it can be found in "Electromagnetism" by Slater and Frank (McGraw Hill 1947) or by searching for it online. In summary, the Biot-Savart law is an empirical law that can be derived from the Ampere-Maxwell equation and Poisson solutions in electromagnetism.
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.
Kipper

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.

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.

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.

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

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

kipper said:
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, ...)

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.

## 1. What is the Biot-Savart Law?

The Biot-Savart Law is a mathematical equation used to calculate the magnetic field produced by a current-carrying wire. It is based on the principle that a magnetic field is created by the motion of electric charges.

## 2. How is the Biot-Savart Law derived?

The Biot-Savart Law is derived from the principle of superposition, which states that the total magnetic field at a point is the sum of the individual magnetic fields produced by each current element in the wire. This derivation involves integrating over the entire length of the wire and considering the direction and magnitude of the current at each point.

## 3. What are the assumptions made when deriving the Biot-Savart Law?

The Biot-Savart Law assumes that the current is constant and that the wire is infinitely long, straight, and thin. It also assumes that the magnetic field is being measured at a distance far enough from the wire that the effects of the wire's thickness can be ignored.

## 4. Why is the Biot-Savart Law important?

The Biot-Savart Law is an essential tool in electromagnetism and is used to calculate the magnetic field in a variety of situations, such as in motors, generators, and transformers. It also helps explain the behavior of magnetic fields in different materials and can be used to predict the behavior of moving charges in a magnetic field.

## 5. Are there any limitations to the Biot-Savart Law?

While the Biot-Savart Law is a powerful tool, it has some limitations. It is only applicable to steady currents and does not take into account the effects of changing electric fields. Additionally, it does not apply to situations where the current is not in a wire, such as in a solenoid or a curved wire. In these cases, more complex equations are needed to calculate the magnetic field.

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