# Derivation of the Biot-Savart Law from Coulomb's Law?

• demoncore
In summary, the Feynman Lectures discuss the idea that a magnetic field can be thought of as a relativistic transformation of an electric field and vice-versa. This is supported by the fact that the magnetic field of a moving point-charge can be transformed away by working in the reference frame of the charge. It is possible to derive the Biot-Savart Law by calculating the Lorentz transformation of the Coulomb force field, as first done by Lorentz himself in 1895. Another approach is to use the relativistic form of the Lorentz force law, which states that the force on a particle in an electric and magnetic field is given by F = q (E + v × B). This approach is
demoncore
The title says it all--I'm working through The Feynman Lectures, and came across the assertion that a magnetic field can be thought of as a relativistic-transformation of an electric field (and vice-versa). This makes sense to me, since the magnetic field of a moving point-charge can easily be transformed away by working in the reference frame of the charge. The electric field seems more (classically) invariant, but I imagine length-contraction/time dilation play a role there. I was wondering if there is a possible derivation of the Biot-Savart Law by calculating the Lorentz transformation of the Coulomb force field, or something along those lines?

Delta2
The answer is yes. In fact, this was actually first done by Lorentz himself in 1895. He derived the Biot-Savart Law from the Lorentz transformation of the Coulomb force field. This derivation is not as straightforward as one might expect, however. It involves some intricate algebraic manipulations and a clever use of vector calculus. Another approach is to use the relativistic form of the Lorentz force law, which states that the force on a particle in an electric and magnetic field is given by:F = q (E + v × B)where q is the charge of the particle, v its velocity and E and B are the electric and magnetic fields respectively. By comparing this equation to the classical form of the Lorentz force law (F = qE + qv × B), one can show that the magnetic field is simply the relativistic transformation of the electric field. This approach is much simpler than Lorentz's original approach.

## What is the Biot-Savart Law?

The Biot-Savart Law is a fundamental equation in electromagnetism that describes the magnetic field produced by a steady current flow. It is named after the French physicist Jean-Baptiste Biot and the French mathematician Félix Savart, who independently discovered the law in the early 19th century.

## What is Coulomb's Law?

Coulomb's Law is a fundamental equation in electrostatics that describes the force between two charged particles. It states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

## How is the Biot-Savart Law derived from Coulomb's Law?

The Biot-Savart Law can be derived from Coulomb's Law by considering a small element of current and using the principles of vector calculus. By integrating over all the elements of the current, the Biot-Savart Law can be obtained in its final form.

## What are the assumptions made in deriving the Biot-Savart Law from Coulomb's Law?

The derivation of the Biot-Savart Law from Coulomb's Law assumes that the current is steady and that the magnetic field is produced by a continuous distribution of current. It also assumes that the magnetic field is generated by a closed loop of current and that the distance between the source of the magnetic field and the point of measurement is much larger than the size of the current loop.

## What are the practical applications of the Biot-Savart Law?

The Biot-Savart Law has many practical applications, including calculating the magnetic fields produced by electric currents in wires, solenoids, and other devices. It is also used in the design and analysis of electric motors, generators, and transformers. Additionally, the Biot-Savart Law is essential in understanding the behavior of magnetic materials and the interaction between magnetic fields and moving charged particles.

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