Magnetic field created by a solenoid

[Pepealej]

Homework Statement

Inside a very long, straight and hollow plastic tube (R << L) we have ionized liquid moving at a velocity v = 100 m/s. The net charge per unit volume is n = 106 electrons/m3. The radius of the tube is R = 10cm. Around the tube we have a solenoid, with n = 100 loops/m, through which circulates a current I = 2 A.

Estimate the magnetic field on the axis of the tube at a distance d >> L.

Book's answer

0 T or, considering the system as a dipole, $$B=\dfrac{\mu_0}{2\pi d^3}N\pi r^2$$

My question

I understand how the magnetic field is 0 T but, how can I consider the system as a dipole to obtain the other answer?

Thanks in advance!

 

[rude man]

Book's answer

0 T or, considering the system as a dipole, $$B=\dfrac{\mu_0}{2\pi d^3}N\pi r^2$$

My question

I understand how the magnetic field is 0 T but, how can I consider the system as a dipole to obtain the other answer?

Thanks in advance!

Well, we can all see B = 0. But their other answer (dipole) makes no sense since neither the ion nor the wire current I is included in their formula. Probably a typo, but ...

I would say:
Consider all the turns of the coil bunched together to form a single-turn loop with current = nLI. That makes using Biot-Savart easy along the axis & is justifiable by d >> L.

As for the ionic current, about all I could see is trying to compute the magnetic vector potential integral A, then B = del x A.