Ampère's force law

MagneticWireAttraction

AmpereDiagram

Ampère's force law describes the interaction between two current-carrying wires and was formulated by André-Marie Ampère in 1820. Ampère's work laid the foundation for the mathematical study of electromagnetism, leading to the development of Maxwell's equations. This law is a pivotal principle in the field of classical electromagnetism, illustrating the force exerted between two parallel current-carrying conductors.

Formulation

Ampère's force law can be expressed as follows: for two infinitesimally small segments of current-carrying wires, the force (\(d\mathbf{F}\)) one segment exerts on the other is directly proportional to the product of their currents (\(I_1\) and \(I_2\)), and the length of the first segment (\(dl_1\)), and inversely proportional to the square of the distance (\(r\)) between them. The direction of the force is perpendicular to the line joining the segments. Mathematically, it is given by:

\[ d\mathbf{F} = k \frac{I_1 I_2 dl_1 dl_2}{r^2} \hat{\mathbf{r}} \]

where \(k\) is a constant of proportionality, and \(\hat{\mathbf{r}}\) is the unit vector from the first to the second segment. In the SI system, the constant \(k\) is often replaced with \(\frac{\mu_0}{4\pi}\), where \(\mu_0\) is the magnetic constant or the permeability of free space, giving the equation a form that directly relates to the magnetic fields produced by the currents.

Applications

Ampère's force law is fundamental in understanding the magnetic interaction between currents. It is used in the design and analysis of electrical machinery, such as motors and generators, and in the study of magnetic materials. The law also underpins the operation of devices like the galvanometer, which measures current through the magnetic forces it produces.

Historical Context

André-Marie Ampère, a French physicist and mathematician, proposed this law in 1820, shortly after the discovery of electromagnetism by Hans Christian Ørsted. Ampère's experiments showed that the magnetic effects produced by currents in wires were analogous to those of magnets, and he was the first to explain these effects in terms of forces between currents.

See Also

• Lorentz force
• Biot-Savart law
• Electromagnetic induction
• Magnetic field