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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/mq / m, possessing different velocities, enters the region of space, where there are uniform parallel electric and magnetic fields with strength EE and induction B,B, at the point OO (see Fig. 3.101 ). The beam direction coincides with the xx axis at the point O.O . A plane screen oriented at right angles to the xx axis is located at a distance ll from the point OO. Find the equation of the trace that the ions leave on the screen. Demonstrate that at z≪lz \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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