Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S7 / 1 / b

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- Solution b

Solution

With zero scalar potential, the nonrelativistic charged-particle Lagrangian in SI units is kinetic energy plus

q A \cdot \dot{r}

. Thus

L = \frac{m}{2} (\overset{x}{˙}^{2} + \overset{y}{˙}^{2} + \overset{z}{˙}^{2}) - \frac{q B _{0}}{2 L} x^{2} \overset{z}{˙} .

The velocity-dependent magnetic term changes canonical momentum but does not contribute to the mechanical energy.

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