Two solenoids are nested coaxially such that their magnetic fields point in oppo

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, 533 turns, and 5.43 A. For the inner solenoid the corresponding quantities are 18.7 cm, 393 turns, and 1.57 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 5.13 mm?

And at what speed, v2, for an orbital radius of 13.7 mm?

Note: this question has two answers but answers units are m/s

Explanation / Answer

For a solenoid magnetic field is given by

B = u0*(N/L)*i

for outer solenoid

Bo = 4*3.14*10^-7*533*5.43/0.219 = 0.0166 T

for inner solenoid

Bi = 4*3.14*10^-7*393*1.57/0.187 = 0.00414 T

Net magnetic field will be

Bnet = Bo - Bi = 0.0166 - 0.00414 = 0.01246 T

Now to find the proton's speed

Fm = Fc

qvB = mv^2/r

v = qBr/m

B = 0.01246 T

q = 1.6*10^-19

r = 5.13*10^-3 m

m = 1.67*10^-27 kg

v = 1.6*10^-19*0.01246*5.13*10^-3/(9.1*10^-31)

v = 1.123*10^7 m/sec

B.

this time r = 13.7*10^-3 m

v = 1.6*10^-19*0.01246*13.7*10^-3/(9.1*10^-31)

v = 3.001*10^7 m/sec

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