Speed of an object in a solenoid

[montoyo]

Hello everyone,
First I do apologize for my terrible English and my noobness at electronics.
I was looking for the formula to compute the speed of an object in a cylindrical solenoid.
This is what I've found:
B = µ0*N*I/L
F = B*I*L
  = µ0*N*I²
 
And according to newton's second law:
a = F/m
  = µ0*N*I²/m

First question: is this formula correct? 
This seems weird because it means that the acceleration of the object inside the solenoid doesn't depend on
the length of the solenoid (L vanishes).

Second question: what is N?
Okay; it's the number of cable loops around the solenoid; but around the whole solenoid?
Or only around the part of the solenoid where the object stands?
Thanks in advance for your help.

[IanB]

The short answer is "zero". The force on an object (such as a steel ball) inside a solenoid is zero.

Regarding your equations, you have:

  B = µ0*N*I/L = µ0*I*(N/L)

The important thing here is that the field intensity depends on the number of turns per unit length. For an "infinitely" long solenoid the field is the same everywhere. If the field is the same everywhere and the object inside it is symmetrical, then the object will experience no force as all magnetic forces will balance out to zero.

Obviously if the solenoid has finite length, then the field will change near the ends of the solenoid and objects here will experience some force due to the field gradient.

Therefore, the answer to your question is more complicated than you think it is, and a simple answer is not possible.

[T3sl4co1l]

The Maxwell stress
s = B^2 / (2*mu_0)
is a pressure like any other, therefore, what causes things to move is a difference.  In the middle of a solenoid, the force is zero, just as the force inside a gas cylinder is zero.  An opening at the end,  where there is a gradient or difference, is what pushes things around.

Tim