# Ampere & Biot-Savart: 2 Questions Answered

• sawer
In summary: I have searched. As I said, some says Ampere discovered the law before Biot-Savart, some says the opposite. There is no clear information about that (at least I can't find)There are two different laws called 'Ampere's law'.The first one, introduced by Ampere in 1823, was for the force between two current circuits. I believe the law of Biot-Savart for the magnetic field of a single current circuit was introduced . The law that is now called 'Ampere's law' for the closed line integral of B must have been derived at some later time. It's derivation required vector techniques that were not known in 1823.

#### sawer

I have 2 questions about those 2 equations.

1-) Which one came first in the history of electromagnetism? Some articles say: Biot-Savart derived their equation from Ampere's Law. Some say: Ampere derived from Biot-Savart. Which one is true?

2-) Since Ampere is special form of Biot-Savart and Biot-Savart is more general form. Why didn't Maxwell chose Biot-Savart instead of Ampere? Why is Ampere's law stated as one of Maxwell equations instead of Biot-Savart?

1. Ampere's Law and Biot-Savart's Law are equivalent in magnetostatics. This means one can be derived from another.
2. I would guess one of Maxwell's reason is mathematical convinience, he liked Faraday's ideas of field lines very much.

zoki85 said:
1. Ampere's Law and Biot-Savart's Law are equivalent in magnetostatics. This means one can be derived from another.
Thank you. I know they are equivalent. But I just want to know which one came first in the history?
zoki85 said:
2. I would guess one of Maxwell's reason is mathematical convinience, he liked Faraday's ideas of field lines very much.
Field lines?
I don't understand the connection between biot-savart, ampere and field lines.

sawer said:
Thank you. I know they are equivalent. But I just want to know which one came first in the history?
Field lines?
I don't understand the connection between biot-savart, ampere and field lines.
I don't which one came first (that you should research by yourself).
Magnetic field lines are closed in space. It is closed line integral (Ampere's) vs line integral (B-S)

zoki85 said:
I don't which one came first (that you should research by yourself).
I have searched. As I said, some says Ampere discovered the law before Biot-Savart, some says the opposite. There is no clear information about that (at least I can't find)

There are two different laws called 'Ampere's law'.
The first one, introduced by Ampere in 1823, was for the force between two current circuits. I believe the law of Biot-Savart for the magnetic field of a single current circuit was introduced . The law that is now called 'Ampere's law' for the closed line integral of B must have been derived at some later time. It's derivation required vector techniques that were not known in 1823.

• vanhees71
sawer said:
Thank you. I know they are equivalent. But I just want to know which one came first in the history?

Field lines?
I don't understand the connection between biot-savart, ampere and field lines.
I must admit I'm not familiar with the early history of electromagnetism. Note that this year we celebrate the 150th anniversary of Maxwell's equations. So there should come up some material about the history of electromagnetism.

Anyway, to answer your questions above. First of all one should note that the fundamental laws from a modern point of view are the differential form of Maxwell's equations for microscopic electrodynamics, i.e., (in Heaviside-Lorentz units which are the most natural units for electromagnetics):
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}=0,$$
$$\vec{\nabla} \times \vec{B} - \frac{1}{c} \partial_t \vec{E}=\frac{1}{c} \vec{j}, \quad \vec{\nabla} \cdot \vec{E}=\rho.$$
Here ##\vec{E}## and ##\vec{B}## are the electric and magnetic components of the electromagnetic field, ##\rho## is the charge density, ##\vec{j}## the current density, and ##c## the speed of light in vacuo.

Ampere's Law and Biot-Savart's Law are now special cases for time-independent fields. Then the electric and magnetic components decouple, and for the magnetic components we deal with the two equations
$$\vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{B}=\frac{1}{c} \vec{j}.$$
The first equation implies that there's a vector potential, which is determined up to the gradient of an arbitrary scalar field. Thus we can introduce the vector potential via
$$\vec{B}=\vec{\nabla} \times \vec{A}$$
and constrain it by an arbitrary "gauge-fixing condition". As we'll see in a moment the most convenient choice is the Coulomb-gauge condition in this case:
$$\vec{\nabla} \cdot \vec{A}=0.$$
Now it is of utmost imporance to keep in mind that from Ampere's law (the 2nd equation above) it necessarily follows that the current must be source-free, i.e.,
$$\vec{\nabla} \cdot \vec{j}=0.$$
Otherwise there's no solution the equation. But now we use the vector potential in Ampere's Law and use the Coulomb-gauge condition, leading to
$$\vec{\nabla} \times (\vec{\nabla} \times \vec{A})=\vec{\nabla} (\vec{\nabla} \cdot \vec{A})-\Delta \vec{A}=-\Delta \vec{A}=\frac{1}{c} \vec{j}.$$
Now we know the solution of this Poisson equation from electrostatics. It's the same equation but now for the three (Cartesian!) vector components of the vector potential rather than the scalar potential in the electrostatic case. Thus you get
$$\vec{A}(\vec{x})=\frac{1}{4 \pi c} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \frac{\vec{j}(\vec{x}')}{|\vec{x}-\vec{x}'|}.$$
Now taking the curl leads to
$$\vec{B}(\vec{x})=\frac{1}{4 \pi c} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \vec{j}(\vec{x}') \frac{\vec{x}-\vec{x}'}{|\vec{x}-\vec{x}'|^3},$$
which is Biot-Savart's Law.

1820 was the year when it all happened. Oersted's published his investigations on the effect of a current-carrying wire on a nearby compass needle.
Ampère, Biot and Savart took up the investigations in France. All did experimental work as well as theoretical. Ampère discovered forces between current-carrying wires among other things. Biot and Savart discovered the inverse proportionality of field strength to distance from a long wire carrying a current. I believe B & S used a ball-ended magnet as probe. Ampère seems always to have seen currents rather than magnets as the primary source of magnetic fields; magnets were explainable in terms of currents – quite an insight! None of this answers your very interesting question, and I haven't any time at the moment to find the answer. Let us know if you do.

sawer said:
2-) Since Ampere is special form of Biot-Savart and Biot-Savart is more general form. Why didn't Maxwell chose Biot-Savart instead of Ampere? Why is Ampere's law stated as one of Maxwell equations instead of Biot-Savart?

Since you mentioned historical interest, one important tidbit is that Ampere's Law, in its original form, was not exactly one of Maxwell's original equations. Maxwell's original set of equations were 20 simultaneous differential equations in 20 variables ( http://rstl.royalsocietypublishing.org/content/155/459.full.pdf+html ). They were reformulated in the vector calculus formalism of Gibbs by Heaviside as the 4 simultaneous equations we are now familiar with in 1884, which played a key part in cementing Gibbs's new vector calculus notation as being useful. Maxwell performed an alternative derivation of Ampere's law, which allowed him to extend its applicability, and imply the existence of electromagnetic waves, the main point of his paper. Biot-Savart is only true for magnetostatics, so it would not be applicable to that situation.

Last edited:
Yes, the key insight by Maxwell was the addition of the "discplacement current", which is of course a misnomer based on some complicated mechanical models Maxwell invented to give a mechanical interpretation of electromagnetic phenomena, which introduced the somewhat unfortunate idea of an ether (or aether, however you prefer to spell it) into physics. It took about 40 years and many brillant physicists to eliminate aether and a purely mechanistical point of view from physics.

Another thing is that from a phenomenological point of view, and classical electromagnetism in matter is a phenomenological model, because you need quantum theory to derive it from microscopic physics, all magnetic phenomena can formally be described as currents. If you have a magnetization you can implment it by a current density (using Heaviside-Lorentz units):
$$\vec{j}_{\text{mag}}=c \vec{\nabla} \times \vec{M}.$$

It's not often one can find fault in a vanHees71 post, but Maxwell certainly didn't introduce the idea of an (a)ether into Physics. Young (c 1803) demonstrated the interference of light, gave a plausible explanation in terms of periodic waves, and even found the range of visible wavelengths. Malus and Fresnel demonstrated polarisation, making it clear that the waves were transverse. Attempts were made, notably by Fresnel in the 1820s, to determine the properties which a mechanical medium (aether) must have in order to allow transverse mechanical waves to propagate at the known speed of light. Two problems were: the medium must surely be very rigid to allow such propagation, so how could planets etc. move freely? Why weren't longitudinal waves detected as well? Others such as Cauchy and Kelvin had a go at devising a mechanical aether. I believe Kelvin's involved cells containing contra-rotating gyroscopically-mounted flywheels!

Maxwell's aether (consisting of vortex cells kept apart from each other by idle-wheels) was unusual and especially powerful as it attempted to account for electric and magnetic phenomena. Maxwell showed that transverse waves should propagate through it at the speed of light. Maxwell is quite clear that he was not putting forward his mechanical ether as a likely candidate for a mode of motion existing in nature, but he seems to have continued to believe that some medium existed to support the propagation of electric and magnetic fields.

• vanhees71
Ok, maybe there were earlier ideas about aether concerning light and optics, but before Faraday and Maxwell it was not clear at all that light is an electromagnetic phenomenon!

Agreed!