question_answer

An electron and a proton are moving on straight parallel paths with same velocity. They enter a semi-infinite region of uniform magnetic field perpendicular to the velocity. Which of the following statement(s) is true?

A) They will never come out of the magnetic field region.

B) They will come out travelling along perpendicular paths.

C) They will come out at the same time.

D) They will come out at different times.

Correct Answer: D

Solution :

Figure shows that the magnetic field \[\vec{B}\] is present on the right hand side of AB. The electron (e) and proton (p) moving on straight parallel paths with the same velocity enter the region of uniform magnetic field. The entry and exit of electron & proton in the magnetic field makes the same angle with AB as shown. Therefore both will come out travelling in parallel paths. The time taken by proton \[{{t}_{p}}=\frac{dis\tan ce}{speed}=\frac{arc}{speed}=\frac{angle\times radius}{speed}=\frac{2\theta \times {{R}_{P}}}{\nu }\] \[=\,\,\,\frac{2\theta }{v}\times \left( \frac{{{m}_{p}}\nu }{eB} \right)=\frac{2\theta {{m}_{p}}}{eB}\] The time taken by electron is \[{{t}_{e}}=\frac{(2\pi -2\theta ){{R}_{e}}}{\nu }=\frac{(2\pi -2\theta )}{\nu }\left( \frac{{{m}_{e}}\nu }{eB} \right)=\frac{(2\pi -2\theta ){{m}_{e}}}{eB}\] clearly \[{{t}_{e}}\] is not equal to \[{{\operatorname{t}}_{p}}\,as\,\,{{m}_{p}}\,>>\,\,{{m}_{e}}\]