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Question:

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

They will come out travelling along parallel paths.

They will come out at different times.

They will never come out of the magnetic field region.

They will come out at the same time.

Solution:

Both the electron and the proton will experience a magnetic force perpendicular to their velocity and the magnetic field. This force will cause them to move in circular paths within the magnetic field region. The radius of the circular path is given by r = mv/qB, where m is the mass, v is the velocity, q is the charge, and B is the magnetic field strength. Since the electron and proton have different masses (the proton is significantly more massive), they will have different radii of curvature. However, because the magnetic force is always perpendicular to the velocity, the paths will be semicircles. After traversing the semicircular path, they will exit the magnetic field region traveling parallel to their initial paths. The time it takes to complete a semicircular path is given by T = πm/qB. Since the mass of the proton is much larger than the mass of the electron, the time taken by the proton to exit the magnetic field region will be much larger than that of the electron. Therefore, they will exit the field traveling parallel paths but at different times.