✎ Lorentz Force Law [The Physics Travel Guide]

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<WRAP lag>$ \vec  F= q \vec E + q\vec v \times \vec B$</WRAP>

====== Lorentz Force Law ======

The Lorentz force law completes classical electromagnetic and describes the effect of electric and magnetic fields on a point charge.

In addition, the [[equations:maxwell_equations|Maxwell equations]] tells us how charges give rise to electric and magnetic fields.

<tabbox Concrete> 
----

**Derivation**

The [[formalisms:lagrangian_formalism|Lagrangian]] for a charge $e$ with mass $m$ in an electromagnetic potential $A$ is
\begin{equation}
\label{eq:Lagrangian-relativistic-EM}
 L(q,\dot{q}) = m\norm{\dot{q}} + eA_i\dot{q}^i 
\end{equation}
so we can work out the Euler--Lagrange equations:
\begin{align*}
 p_i = \frac{\pa L}{\pa\dot{q}^i} &= m\frac{\dot{q}_i}{\norm{\dot{q}}} + eA_i\\
 &= m v_i + e \,A_i
\end{align*}
where $v$ is the velocity, which we normalize such that $\norm{v}=1$.  An important observation is that here momentum is no longer simply mass times velocity! Continuing the analysis, we find the force
\begin{align*}
 F_i = \frac{\pa L}{\pa q^i} &= \frac{\pa}{\pa q^i}\Bigl(e\,A_j\dot{q}^j\Bigr)\\
 &= e\frac{\pa A_j}{\pa q^i} \dot{q}^j
\end{align*}
So the [[equations:euler_lagrange_equations|Euler-Lagrange equations]] give us (using $A_i=A_j\Bigl(q(t)\Bigr)$:
\begin{align*}
 \dot{p} &= F \\
 \frac{d}{dt}\Bigl(mv_i+eA_i\Bigr) &= e\frac{\pa A_j}{\pa q^i}\dot{q}^j\\
 m\frac{d v_i}{dt} &= e\frac{\pa A_j}{\pa q^i}\dot{q}^j - e\frac{d A_i}{dt}\\
 m\frac{d v_i}{dt} &= e\frac{\pa A_j}{\pa q^i}\dot{q}^j 
 - e\frac{\pa A_i}{\pa q^j}\dot{q}^j\\
 &= e\left(\frac{\pa A_j}{\pa q^i} - \frac{\pa A_i}{\pa q^j}\right)\dot{q}^j .
\end{align*}
Here, term in parentheses is $\fvect{F}_{ij}=$ the electromagnetic field, $F=dA$.  Therefore, the equations of motion are

\begin{equation}
\boxed{
 \eqngapabove \gapleft
 m\frac{d v_i}{dt} = e\fvect{F}_{ij}\dot{q}^j, \quad\text{(Lorentz force law)}
 \gapright \eqngapbelow
}
\end{equation}

which we call the Lorentz law.)
 
<tabbox Abstract>

<blockquote>The classical mechanics of an electron propagating in an electromagnetic field on a spacetime X is all encoded
in a differential 2-form on X, called the Faraday tensor F, which encodes the classical Lorentz force that the
electromagnetic field exerts on the electron.<cite>https://arxiv.org/abs/1601.05956</cite></blockquote>

For a derivation, using the Ehrenfest theorem, see http://www.physics.drexel.edu/~bob/PHYS517/Ehrenfest.pdf

</tabbox>

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