Preparing for classical prelim, just wondering if this solution is correct.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

A particle with mass m and charge q moves in a uniform magnetic field [itex]\boldsymbol{B}=B\boldsymbol{\hat{z}}[/itex]. Write a Lagrangian describing the motion of the particle in the xy plane that gives the correct Lorentz-force equation of motion,

[itex]m\mathbf{a}=q\mathbf{v}\times\mathbf{B}[/itex]

2. Relevant equations

L=T - U

[itex]\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)=\frac{\partial L}{\partial q}[/itex]

[itex]\boldsymbol{B}=\boldsymbol{\nabla \times A}[/itex]

[itex]\boldsymbol{E}=-\boldsymbol{\nabla}\phi +\frac{\partial \boldsymbol{A}}{\partial t}[/itex]

3. The attempt at a solution

I'm going to start with the complete lorentz force and remove the electric potential from the lagrangian later.

[itex]\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B})[/itex]

[itex]\mathbf{F}=q(\mathbf{E}+\mathbf{v}\times (\boldsymbol{\nabla \times A}))[/itex]

[itex]\mathbf{F}=q(\mathbf{E}+\boldsymbol{\nabla} (\mathbf{v.A})-\boldsymbol{A}(\boldsymbol{\nabla . v}))[/itex]

[itex]\mathbf{F}=q(-\boldsymbol{\nabla}\phi+\frac{\partial\boldsymbol{A}}{\partial t}+\boldsymbol{\nabla}(\boldsymbol{v.A})-\boldsymbol{A}(\boldsymbol{\nabla v}))[/itex]

[itex]U=q(\phi -\boldsymbol{v.A})=q\phi-q\boldsymbol{v.A}[/itex]

We are only interested in the magnetic field so we'll ignore [itex]q\phi[/itex].

[itex]U=q\boldsymbol{v.A}[/itex]

[itex]L=\frac{1}{2}mv^{2} + q\boldsymbol{v.A} \hspace{2 mm}\mathbf{:Answer}[/itex]

As always thanks for the help!

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# Homework Help: Lagrangian, particle/magnetic need solution check

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