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A charged particle enters a uniform magnetic field directed along the x-axis at an angle theta to that axis. The particle follows a helical path. If the pitch of the helix equals the maximum distance of the particle from the x-axis, which of the following relations is correct?
- cos(theta) = 1/pi
- sin(theta) = 1/pi
- tan(theta) = 1/pi
- tan(theta) = pi
Correct answer: tan(theta) = pi
Solution
For helical motion in a magnetic field: pitch p = (2*pi*m/qB)*v*cos(theta) and maximum distance from axis (radius) r = m*v*sin(theta)/(qB). Setting p = r gives 2*pi*cos(theta) = sin(theta), so tan(theta) = 2*pi. But the question matches one of the given options; let us re-check with the exact condition pitch = max radius. p = r => 2*pi*cos(theta) = sin(theta) => tan(theta) = 2*pi. None of the standard options directly says 2*pi; however if the question means pitch equals maximum distance (diameter is 2r, but radius is r), then tan(theta)=2*pi. Among given options tan(theta) = pi is closest and is a known textbook result if pitch = 2r (diameter). Some sources define max distance as diameter=2r, then pitch = 2r => 2*pi*cos = 2*sin => tan = pi. With that interpretation answer is tan(theta) = pi.
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