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Homework Statement
Consider a particle of mass m and charge q that moves in an E-field [itex]\vec{E}=\frac{E_0}{r}\hat{r}[/itex] and a uniform magnetic field [itex]\vec{B}=B_0\hat{k}[/itex]. Find the scalar potential and show the vector potential is given by [itex]\vec{A}=\frac{1}{2}B_0 r \hat{\theta}[/itex]. Then obtain the Lagrange equations of motion and identify the conserved quantities
Homework Equations
Lagrange equations
The Attempt at a Solution
Using cylindrical coords,
[tex]L=T-V=\frac{1}{2}m \left ( \dot{r}^2+\dot{\theta}^2+\dot{z}^2 \right) - e\phi + e\vec{v} \cdot \vec{A}[/tex]
[tex]L = \frac{1}{2}m \left ( \dot{r}^2+\dot{\theta}^2+\dot{z}^2 \right) - eE_0\ln(r) + \frac{1}{2}e\dot{\theta}B_0r[/tex]
Using the Lagrange equation,
[tex]0 = m\ddot{r} + eE_0\frac{1}{r} - \frac{1}{2}e\dot{\theta}B_0[/tex]
[tex]0=m\ddot{\theta}[/tex]
[tex]0=m\ddot{z}[/itex]
Correct?