Draft:Magnetic force

In electromagnetism, Magnetic force arises from moving electric charges (i.e Lorentz force), particularly charges that are not in static state in any condition. In classical physics, a magnetic field B is defined by the force it exerts on a moving charge. Specifically, a charge q moving with velocity v in a magnetic field experiences a Lorentz force i.e.$F=q.v\times B$, always perpendicular to v and B. While, Richard Feynman emphasized that “one part of the force between moving charges we call the magnetic force." It is really one aspect of an electrical effect. In other words, magnetism naturally emerges when charges move, and via special relativity the electric and magnetic fields are interrelated. Faraday introduced the modern field concept (1831), showing that changing currents produce magnetic fields that in turn induce electric effects, while this phenomena is termed as electromagnetic effect.

Definition

Magnetic force arises from what is known as magnetic field, which is the physical area surrounding a magnetic element (e.g. an electric current, dipole magnet etc.). Magnetic field is defined is an infinitely long field arising from an electric current that spans in the physical space around that point to infinitely, though the natural effect of this field gets diminished as the field grows longer or the distance between field at any point vs the location of current increases, defined by the mathematical formula:

$B=\frac{{\mathrm{\mu }}_{\mathrm{0}}I}{\mathrm{2}\mathrm{\pi }r}$

where,

B: Magnetic field; $\mu$: Permeability of free space; I: current; 2$\pi$r: area of the conductor carrying the current.

A magnetic force does no work on the particle since the velocity of the charged particle and the magnetic force are always perpendicular to each other.Particularly, magnetic fields have no monopoles: Gauss’s law for magnetism is, meaning field lines form continuous loops. In special relativity, electric and magnetic fields mix between frames, but classically one can view B as the force field felt by moving charges, governed by the right-hand-rule cross product.

Force on moving charge and current

A point charge q moving with velocity v and having a magnetic field of B, have magnetic force:

$F=q.v\times B$

and its magnitude is

$|F|=qvBsin\theta$

where $\theta$ is the angle between the v and B. In uniform B ⟂ v, the particle undergoes circular motion with radius,$r=\frac{mv}{|q|B}$ (since $qvB=\frac{m{v}^2}{r}$ for centripetal force). If v has a component parallel to B, the motion is helical.

For a current-carrying wire, consider many charges moving through a conductor. If the wire has charge carrier density n and cross-sectional area A, its current is $I=qnvA$. The total force on a straight segment of wire of length L in a uniform field is.

$F=I.L\times B$

pointing perpendicular to both L and B. Equivalently, the force per unit length is $F/L=I.Bsin\theta$ where $\theta$ is the angle between the wire and magnetic field B. In practice, one often writes $F=ILB$ (for perpendicular geometry). For example, A proton (q =$1.6\times 10^{-19}C$, m = $1.6\times 10^{-27}kg$ moving at v = $10^{6}m/s$ perpendicular to a 1 T field experiences $F=qvB\approx 1.6\times 10^{-12}N$ . Its circular orbit radius is r = $10^{-}5m$.

Mathematical formulation

The magnetic field B produced by steady currents can be computed by the Biot–Savart law. For an infinitesimal current element the contribution to the magnetic field at point P is:

$B=\frac{{\mu }_0}{4\pi }\frac{Idl\times \hat{r}}{r^2}$

where $\hat{r}$ is the unit vector from the element to P and r the distance. Integrating over a wire yields:

$B=\frac{{\mu }_0}{4\pi }\int \frac{Idl\times \hat{r}}{r^2}$

For example, an infinitely long straight wire carrying current I has the known result $B=\frac{{\mathrm{\mu }}_{\mathrm{0}}I}{\mathrm{2}\mathrm{\pi }r}$ in circular loops around the wire (by integrating Biot–Savart or using Ampère’s law). More generally, Ampère’s circuital law (with Maxwell’s correction) states:

$\oint B.dS={\mu }_0{I}_{enc}+{\mu }_0{ϵ}_0\frac{d{\varphi }_E}{dT},I$

so that in steady currents $\oint B.dS={\mu }_0{I}_{enc}$.

Maxwell’s equations summarize all of classical electromagnetism. In addition to the above, Gauss’s law for magnetism $\varphi =\frac{q_{enc}}{{ϵ}_0}$, and Faraday’s law $ϵ=-\frac{d{\varphi }_0}{dt}$. These equations and the Lorentz law provide one the complete description of electric and magnetic fields and forces.

Historical development

1. 1820 – Ørsted’s discovery: In 1820, Hans Ørsted found that a current through a wire deflects a compass needle, showing that electric currents create magnetic fields.
2. 1821 – Ampère’s electrodynamics: In 1821, André-Marie Ampère established that parallel currents attract or repel, formulating the fundamental laws of forces between current elements.
3. 1831 – Faraday’s induction: In 1831, Michael Faraday discovered electromagnetic induction (moving magnets and coils induce currents) and defined field concepts, laying the groundwork for electric motors and generators.
4. 1873 – Maxwell’s unification: In 1873, James Clerk Maxwell synthesized previous laws into four equations (Maxwell’s equations) relating E and B, and predicted electromagnetism. Maxwell’s theory showed that light is an electromagnetic wave, uniting optics with electromagnetism.
5. (Later, 1887 – Hertz confirmed Maxwell by creating radio waves.)

However in later periods the contributions in electromagnetism were by few more notables such that many other scientists contributed (e.g. Coulomb’s force law, Gauss’s exploration of fields), but Ørsted, Ampère, Faraday, and Maxwell were seem to be central figures in the development of magnetic force theory.