Does Solenoid Diameter Affect Acceleration of a Metal BB?

[joe_cool2]

Homework Statement

We're given two solenoids with identical currents and turns per unit length. But the solenoids have different diameters. Would a metal BB experience a larger acceleration in the smaller diameter tube, assuming we use the same BB in both solenoids it accelerates along the z-axis, which goes through the center of the solenoid?

Homework Equations

Magnetic field through a solenoid:

$B = \mu_0NI$

Where N is number of turns per unit length and I is the current.

Biot-Savart Law:

$$B = \frac{1}{4\pi}\frac{I\Delta \vec{s} \times \vec{r}}{r^2}$$

The Attempt at a Solution

My thought is that since the BB experiences the same uniform magnetic field in both solenoids, the acceleration is the same. The equation for the magnetic field through a solenoid does not include diameter, I notice. But I was having a little debate on this subject and my friend says he would expect the field to sort of dissipate as the diameter increases. This would make total sense if it were a point charge; you'd use Biot-Savart's law. Who is right?

 

[tiny-tim]

hi joe_cool2!

(what's a BB ? )

The equation for the magnetic field through a solenoid does not include diameter, I notice. But I was having a little debate on this subject and my friend says he would expect the field to sort of dissipate as the diameter increases.

you are correct …

if the solenoid is infinitely long, the magnetic field is uniform

if the solenoid is ordinary long (ie, a lot longer than wide), the magnetic field is nearly uniform, and you can assume this to be so in exam questions!

from hyperphysics …

"The magnetic field is concentrated into a nearly uniform field in the centre of a long solenoid"​

 

[TSny]

See here for description of BBs. They are often made of steel with a copper or zinc coating, lead, or plastic. I don't see how a BB in a B field (pun intended, sorry) would experience any noticeable acceleration.