In a three-dimensional Cartesian coordinate system (x,y,z), an electron (mass m, charge −e, e>0) is accelerated from rest through a potential difference V and injected from the origin (0,0,0) in the +x direction with initial speed v0. Gravity and relativistic effects are negligible.
Step 1: Velocity Selector In the region 0≤x≤L, a uniform electric field E=(0,E0,0) and a uniform magnetic field B=(0,0,B0) are applied simultaneously. The injected electron travels straight along the x-axis without any deflection.
Step 2: Magnetic Deflection The electric field is turned off (E=0), while the magnetic field B remains. The same electron with initial speed v0 is injected again from the origin. Inside 0≤x≤L, the electron is deflected by the Lorentz force and follows a curved path in the xy-plane, exiting the magnetic field region through the plane x=L.
Step 3: Drift Space The region x>L is a field-free drift space. The electron travels in a straight line and hits a fluorescent screen (parallel to the yz-plane) placed at x=L+D. The y-coordinate of the impact point on the screen is recorded as displacement Y.
The instrument is precisely set up with drift length D=3L. The observed displacement on the screen is exactly Y=(3−3)L.
From these results, find the specific charge e/m of the electron.
Important: Do not use any small-angle approximation (such as sinθ≈θ or y≈L2/(2R)). Derive the result using exact geometric relations.
The specific charge e/m can be written as A×1011C/kg for some real number A.
Compute 100A and give the answer as a positive integer.