Question

PLZ SHOW FULLL WORK I NEED HELP ASAP

A long solenoid with diameter 4.0 cm has 1000 turns uniformly distributed over a length of 50.0 cm. The solenoid is in a vacuum, and a lithium nucleus (4 neutrons and 3 protons) is in a clockwise circular orbit inside the solenoid as shown in the figure. It takes 50 ns (50 X 10'9 s) for the lithium nucleus to complete one orbit. What is the magnitude and direction of the magnetic field inside the solenoid? (Ans: B = 3.05 T) What is the magnitude and direction of the current in the solenoid. (Ans: I = 1.2 X 103 A) Apply Ampere's law, ^B ds = jyl, to evaluate the line integral ^B ds for the following

Answer 1

1) a) the forces are balanced so,

mv^2/r = qvB

==> mv/r = qB

==> B = mv/qr

v = 2pi*r/t

so B = m*2pi*r/q*r*t = m*2pi/qt

==> B = 7*1.67*10^-27*2*pi/(3*1.6*10^-19*50*10^-9) = 3.05 T

b) B = mu0*N*I/L

==> I = B*L/mu0*N

==> I = 3.05*0.5/(4*pi*10^-7*100) = 1.2*10^3 A

direction CCW

2) a) $\oint B.ds = \mu_{0} I_{enclosed} \newline \oint B.ds = \mu_{0} (I_2-I_1) \newline \oint B.ds = 4*pi*10^{-7}*(4-2) = 2.48*10^-6$

b) $\oint B.ds = \mu_{0} I_{enclosed} \newline \oint B.ds = \mu_{0} (I_5+I_6-I_4) \newline \oint B.ds = 4*pi*10^{-7}*(3+8-5) =7.44*10^-6$