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Motion of Charged Particles in Electric and Magnetic Fields

Problem 3.395

A charged particle with specific charge \(q / m\) starts moving in the region of space where there are uniform mutually perpendicular electric and magnetic fields. The magnetic field is constant and has an induction \(B\) while the strength of the electric field varies with time as \(E=E_{m} \cos \omega t,\) where \(\omega=q B / m .\) For the non-relativistic case find the law of motion \(x(t)\) and \(y(t)\) of the particle if at the moment \(t=0\) it was located at the point \(O\) (see Fig. 3.104 ). What is the approximate shape of the trajectory of the particle?

Reveal Answer
 3.395. y=a2ωtsin⁡ωt,x=a2ω2(sin⁡ωt−ωtcos⁡ωt), where a=qEm/m. The trajectory has the form of unwinding spiral. \begin{aligned} &\text { 3.395. } y=\frac{a}{2 \omega} t \sin \omega t, \quad x=\frac{a}{2 \omega^{2}}(\sin \omega t-\omega t \cos \omega t), \quad \text { where }\\ &a=q E_{m} / m . \text { The trajectory has the form of unwinding spiral. } \end{aligned}
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