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THIS IS ONE QUESTION WITH MULTIPLE PARTS!

The circular motion of charges in a constant magnetic field has many applications, from laboratory experiments to microwave ovens. Consider one of the scientific applications of this "cyclotron motion", the Mass Spectrometer.

In the mass spectrometer, atoms (the small circles) of mass m are positively charged by removing one electron ( = +1.6 10 19 ). They are accelerated by an electric field created by a voltage drop (electric potential change) . The moving atoms enter a region of constant magnetic field, , so their trajectory is a quarter of a circle of radius . If the field and the potential are correct, they are detected in a detector that measures their charge.

The electric field is zero outside of the acceleration region, , and the magnetic field is constant in the gray region in the diagram and is zero outside. To accelerate the positively charged atoms (increase their kinetic energy), < 0 in the direction of motion.

A) What direction is the constant magnetic field , into the page or out of the page? Explain. Your explanation should include the direction of the force at the point P.

B) Using the centripetal acceleration of an atom moving in a circle of radius , a = (^2)/, derive an equation for the speed in terms of , , and other physical quantities.

C) The speed of the atom is determined by . Assume the atoms are initially at rest. Derive an equation for the speed in terms of and other physical quantities.

D) Using the results from (B) and (C), derive an equation for the potential that will result in the atom traveling in a circle of radius , in terms of , , and other physical quantities.

E) Explain the physics of your result from Part D. How would you need to change | | (larger, smaller, no change) if you wanted a smaller for a fixed , , and ? How would you need to change || if is larger while , , and stay the same? Explain using the physics of the electric and magnetic forces, energy, etc.

F) mass spectrometer measures masses by determining the needed to hit the detector. Let = 0.20 and = 0.050 . i. What is needed to detect carbon atoms, = 2.0 10 26 ? ii. What is needed to detect oxygen atoms, = 2.67 10 26 ?