Area Constant (k_a)

[pmbov]

Hello all,
I am trying to find the acceleration of the closing of an armature on a solenoid driven injector. I am using the reference Magnetic Actuators and Sensors, 2nd edition by John R. Brauer. The author states the equation for acceleration equation is:

a=P*(pi*(k_a)*(R_p)^2)/(density of material)*(volume of armature); simply, a = force/mass

Where:
P = magnetic pressure
k_a = Area constant <= 1
R_p = radius of the armature

The author states that k_a is an area constant but does not explain what this number could be or even where to find it. Can anyone please help guide me where I can find this constant or even how to calculate it?

 

[MagBen]

it seems the square of (radius of armature/radius of the coil)

 

[MagBen]

The radius difference may not matter, when armature is smaller than coil radius, the area in between can be ignored since it does not take a part in the energy transfer that will follow.
so K_a maybe only (1/mu0 - 1/mu armature), when permeability of armature is much larger than air (=1), then it = 1.