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formulas:lorentz_force_law [2017/10/27 13:12]
jakobadmin ↷ Page moved from classical_theories:electrodynamics:lorentz_force_law to theories:classical_theories:electrodynamics:lorentz_force_law
formulas:lorentz_force_law [2018/05/13 09:18] (current)
jakobadmin ↷ Page moved from equations:lorentz_force_law to formulas:lorentz_force_law
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-====== Lorentz Force Law ======+<WRAP lag>$ \vec  Fq \vec E + q\vec v \times \vec B$</​WRAP>​
  
-<tabbox Why is it interesting?> ​ 
  
-<tabbox Layman> ​+====== Lorentz Force Law ======
  
-<note tip> 
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. 
-</​note>​ 
-  ​ 
-<tabbox Student> ​ 
  
-<note tip> +<​tabbox ​Intuitive
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas. +
-</​note>​ +
-  +
-<​tabbox ​Researcher+
  
-For derivation, using the Ehrenfest theorem, see http://​www.physics.drexel.edu/​~bob/​PHYS517/​Ehrenfest.pdf+The Lorentz force law completes classical electromagnetic and describes the effect of electric and magnetic fields on point charge
  
---> Common Question 1#+In addition, the [[equations:​maxwell_equations|Maxwell equations]] tells us how charges give rise to electric and magnetic fields.
  
-  
-<-- 
  
---> Common Question 2# 
  
-  
-<-- 
   ​   ​
-<​tabbox ​Examples+<​tabbox ​Concrete 
 +**Derivation**
  
---> Example1#+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|{\dot{q}}| + eA_i\dot{q}^i  
 +\end{equation} 
 +so we can work out the Euler--Lagrange equations:​ 
 +\begin{align*} 
 + p_i = \frac{\partial L}{\partial \dot{q}^i} &= m\frac{\dot{q}_i}{|{\dot{q}}|} + eA_i\\ 
 + &​= m v_i + e \,A_i 
 +\end{align*} 
 +where $v$ is the velocity, which we normalize such that $|v|=1$. ​ 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{\partial L}{\partial q^i} &= \frac{\partial}{\partial q^i}\Bigl(e\,​A_j\dot{q}^j\Bigr)\\ 
 + &​= e\frac{\partial A_j}{\partial 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{\partial A_j}{\partial q^i}\dot{q}^j\\ 
 + ​m\frac{d v_i}{dt} &= e\frac{\partial A_j}{\partial q^i}\dot{q}^j - e\frac{d A_i}{dt}\\ 
 + ​m\frac{d v_i}{dt} &= e\frac{\partial A_j}{\partial q^i}\dot{q}^j  
 + - e\frac{\partial A_i}{\partial q^j}\dot{q}^j\\ 
 + &​= e\left(\frac{\partial A_j}{\partial q^i} - \frac{\partial A_i}{\partial q^j}\right)\dot{q}^j . 
 +\end{align*} 
 +Here, term in parentheses is $F_{ij}=$ the electromagnetic field, $F=dA$. ​ Therefore, the equations of motion are
  
 +\begin{equation}
 + ​m\frac{d v_i}{dt} = e F_{ij}\dot{q}^j,​
 +\end{equation}
 +
 +which we call the Lorentz law.
    
-<--+<tabbox Abstract> ​
  
---Example2:#+<​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 Why is it interesting?> ​
  
-  
-<-- 
-  ​ 
-<tabbox History> ​ 
  
 </​tabbox>​ </​tabbox>​
  
  
formulas/lorentz_force_law.1509102749.txt.gz · Last modified: 2017/12/04 08:01 (external edit)