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formulas:lorentz_force_law [2018/03/28 13:10]
jakobadmin
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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 +<WRAP lag>$ \vec  F= q \vec E + q\vec v \times \vec B$</​WRAP>​
 +
 +
 ====== Lorentz Force Law ====== ====== Lorentz Force Law ======
  
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 <tabbox Intuitive> ​ <tabbox Intuitive> ​
  
-<note tip> +The Lorentz force law completes classical electromagnetic and describes the effect of electric and magnetic fields on a point charge. ​ 
-Explanations in this section should contain no formulasbut instead colloquial things like you would hear them during a coffee break or at a cocktail party+ 
-</​note>​+In additionthe [[equations:​maxwell_equations|Maxwell equations]] tells us how charges give rise to electric and magnetic fields
 + 
 + 
   ​   ​
 <tabbox Concrete> ​ <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|{\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}
  
-<note tip> +which we call the Lorentz law.
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas. +
-</​note>​+
    
 <tabbox Abstract> ​ <tabbox Abstract> ​
formulas/lorentz_force_law.1522235428.txt.gz · Last modified: 2018/03/28 11:10 (external edit)