Question

Two solenoids are nested coaxially such that their magnetic fields point in opposite directions. Treat the solenoids as ideal. The outer one has a radius of 20 mm, and the radius of the inner solenoid is 10 mm. The length, number of turns, and current of the outer solenoid are, respectively, 21.9 cm, 519 turns, and 5.13 A. For the inner solenoid the corresponding quantities are 18.7 cm, 355 turns, and 1.77 A.

A) At what speed, v1, should a proton be traveling, inside the apparatus and perpendicular to the magnetic field, if it is to orbit the axis of the solenoids at a radius of 6.71 mm?

B And at what speed, v2, for an orbital radius of 17.9 mm?

Answer 1

B=mu_0(N/L)I where B is magnetic field, N is # of turns, L is length, and I is current. Calculate the magnetic fields for the two solenoids. Take the difference of magnetic fields to get the superposition of magnetic fields (since they point in opposite directions). You now have the net magnetic field.

Now you need to relate radius of curvature to magnetic field. The force the proton feels is determined by the Lorentz force (in this case, F=q*VxB, where F is force, q is positive for proton, and V is velocity). This Lorentz force will cause the proton to travel in a circle. Because of centripetal acceleration, F=ma=m*V^2/r, where m is mass of proton and r is radius of curvature.

From this, we get F=q*VxB=m*V^2/r

soooo

we know q (look up q for proton online), B (you calculate that), m (look it up on google), and r (given)

V=q*B*r/m (you can ignore the cross product because the path of the proton is always perpendicular to the magnetic field)