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Motion of Charged Particles in Electric and Magnetic Fields

Problem 3.388

A narrow beam of identical ions with specific charge \(q / m\), possessing different velocities, enters the region of space, where there are uniform parallel electric and magnetic fields with strength \(E\) and induction \(B,\) at the point \(O\) (see Fig. 3.101 ). The beam direction coincides with the \(x\) axis at the point \(O .\) A plane screen oriented at right angles to the \(x\) axis is located at a distance \(l\) from the point \(O\). Find the equation of the trace that the ions leave on the screen. Demonstrate that at \(z \ll l\) it is the equation of a parabola.

Reveal Answer
 3.388. z=ltan⁡qB22mEy; for z≪1 this equation reduces to y=(2mE/ql2B2)z2\begin{aligned} &\text { 3.388. } z=l \tan \sqrt{\frac{q B^{2}}{2 m E}} y ; \text { for } z \ll 1 \text { this equation reduces to }\\ &y=\left(2 m E / q l^{2} B^{2}\right) z^{2} \end{aligned}
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