# Why is force on moving charges in magnetic field perpendicular?

(1)The magnetic field around a conductor where there's a current can easily be demonstrated using something like a compass needle. The magnetic field lines will be perpendicular to the direction of the current.

Now, charges will not be affected by a electromagnetic force unless they are moving. However, in my textbook they say when the velocity is perpendicular to some magnetic field they are in, a force will be perpendicular to the charges - direction given by right-hand rule.

Obviously that can as well be demonstrated very easily through experiments. But is there a way to show this theoretically with magnetic field lines, i.e. why the force will be directed upwards in some cases for example?

• There is no answer to any "why" question in physics: it is so because it is so (it could have been otherwise if the universe had other fundamental rules). Physics describes "how" things work, not "why". – gented Jul 20 '17 at 8:46
• How did you define a magnetic field $B$ if not as the field such that $F=v\times B$? – ACuriousMind Jul 20 '17 at 8:54

The force on a current carry conductor is given by the equation to find magnitude as given below and its direction given by Flemings Left Hand Rule: $$F=BIlsin(\theta)$$ This formula can be derived from the force on a moving charge $F=Bvqsin(\theta)$ or vice versa where $v$ is the velocity of charge. See, current in a wire acts like a moving charge and moving charges generate a circular magnetic field around them( details on their magnitude can be found using Biot-Savart law) so the resultant force due to field of moving charge or current becomes perpendicular to the direction of motion/flow of current. Below is an image showing the fields. Given that there is an electric force ${\bf F}_e=q{\bf E}$, the existence of a magnetic force of the form ${\bf F}_m=q{\bf v}\times{\bf B}$ may be deduced by requiring Lorentz invariance.
Both expressions come from a single relativistically covariant expression for the four-force, $f_\mu=q\dot x^\nu F_{\mu\nu}$, where $\dot x^\mu\equiv dx^\mu/d\tau$ is the four-velocity and $F_{\mu\nu}$ is the electromagnetic field strength, whose components can be found in basic electrodynamics textbooks. In particular, the spatial components ($\mu=1,2,3$) give rise to the combined electric+magnetic force, $F_i=q(E_i+\epsilon_{ijk}v_jB_k)$, or, in vector notation, ${\bf F}=q({\bf E}+{\bf v}\times{\bf B})$.