The physics
A solenoid is a long coil of many turns. Near its middle, the field inside is uniform and runs along the axis.
B = μ0 n I n = N / L (turns per metre) μ0 = 4π × 10⁻⁷ T m/A
Walk a rectangle with one side inside, along the axis. Only that side adds to ∮B·dl, so B l = μ0 (n l) I.
At an end, all the turns lie on one side of the probe. There B is half the inside value. The magnetic intensity H leaves out the μ0.
Bend = μ0 n I / 2 H = n I (A/m)
An electron moving across the axis goes round in a circle. The time for one turn does not depend on its speed. An electron moving along the axis feels no force.
T = 2πm / eB