Paper 4, Section I, B

Dynamics and Relativity | Part IA, 2010

A particle of mass mm and charge qq moves with trajectory r(t)\mathbf{r}(t) in a constant magnetic field B=Bz^\mathbf{B}=B \hat{\mathbf{z}}. Write down the Lorentz force on the particle and use Newton's Second Law to deduce that

r˙ωr×z^=c\dot{\mathbf{r}}-\omega \mathbf{r} \times \hat{\mathbf{z}}=\mathbf{c}

where c\mathbf{c} is a constant vector and ω\omega is to be determined. Find c\mathbf{c} and hence r(t)\mathbf{r}(t) for the initial conditions

r(0)=ax^ and r˙(0)=uy^+vz^\mathbf{r}(0)=a \hat{\mathbf{x}} \quad \text { and } \quad \dot{\mathbf{r}}(0)=u \hat{\mathbf{y}}+v \hat{\mathbf{z}}

where a,ua, u and vv are constants. Sketch the particle's trajectory in the case aω+u=0a \omega+u=0.

[Unit vectors x^,y^,z^\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}} correspond to a set of Cartesian coordinates. ]

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