Two solenoids are nested coaxially such that their magnetic

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, 21.1 cm, 535 turns, and 4.67 A For the inner solenoid the corresponding quantities are 19.3 cm, 369 turns, and 1.55 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 5.49 mm? And at what speed, v_2, for an orbital radius of 14.9 mm? How do the particle\'s charge, the orbital radius, the magnetic field strength, and partide\'s mass relate to the particle\'s speed? How do a solenoid\'s length, number of turns, and current relate to the strength of the magnetic field inside the solenoid? What is magnetic field strength inside the inner solenoid and between the solenoids? The elementary charge is 1.602 Times 10^-19 C, the mass of the proton is 1.673 Times 10^-27 kg, and the vacuum permeability constant has the value 4pi Times 10^-7 T.m/A.

Solution

Magnetic field due to a solenoid = u0 (N / L ) I

Bnet = B1 - B2 = u0 [( N1 I1 / L1) - (N2 I2 / L2)]

magnetic force always act in perpendicular direction of direction of velocity.

hence charge will move in circular path,

magnetic force = m v^2 / r

q v B = m v^2 / r

v= q r B / m

v = q r u0 [( N1 I1 / L1) - (N2 I2 / L2)] / m

for v1:

Bnet = (4pi x 10^-7) [ (535 x 4.67 /0.211) - (369 x 1.55 / 0.193) ]

B = 0.0112

v1 =(1.6 x 10^-19 x 5.49 x 10^-3 x 0.0112 ) / (1.673 x 10^-27 )

v1 = 5880.5 m/s

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for v2 :

v2 =(1.6 x 10^-19 x 14.9 x 10^-3 x 0.0112 ) / (1.673 x 10^-27 )

v2 = 15959.83 m/s