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, 20.9 cm, 551 turns, and 5.93 A. For the inner solenoid the corresponding quantities are 18.9 cm, 329 turns, and 1.83 A. At what speed, v_1, 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 5.07 mm? And at what speed, v_2, for an orbital radius of 14.3 mm?

Explanation / Answer

for outer solenoid

L1 = 0.209 m

r1 = 20 mm = 0.02 m

n1 = 551

i1 = 5.93 A

B1 = uo*(n1/L1)*i1 = 4*pi*10^-7*(551/0.209)*5.93 = 0.01965 T

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for inner solenoid

L2 = 0.189 m

r2 = 10 mm = 0.01 m

n2 = 329

i2 = 1.83 A

B2 = uo*(n2/L2)*i2 = 4*pi*10^-7*(329/0.189)*1.83 = 0.004 T

Bnet = B1 - B2 = 0.01565 T

for circular motion

v = qBR/m

v = (1.6*10^-19*0.01565*5.07*10^-3)/(1.67*10^-27)

V1 = 7602 m/s

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R = 14.3 > r2

Bnet = B1

v2 = (1.6*10^-19*0.01965*14.3*10^-3)/(1.67*10^-27)

v2 = 26921.7 m/s

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