Solenoids are springshaped coils of wire commonly used in electromagnets. If you run an electric current through a solenoid, a magnetic field will be generated. The magnetic field can exert a force on charged particles that is proportional to its strength. To calculate the force from a solenoid's magnetic field, you can use this equation:
Force = charge x velocity of the charge x magnetic field strength
As you can see from the equation, to calculate force we first need to know the magnetic field strength, which is dependent on the characteristics of the solenoid. We can substitute these parameters into the force equation get:
Force = charge x velocity of the charge x (magnetic constant x number of turns in solenoid x current)
The calculation looks complicated, but really it's just multiplying a bunch of measurable variables together.

Write the equation for the force that a solenoidal electromagnet will exert on a passing charge:
Force = Q x V x (magnetic constant x N x I)
Q = charge of passing point charge
V = velocity of point chart
magnetic constant = 4 x pi x 10^7 (reference 3)
N = number of turns in solenoid
I = current running through solenoid 
Determine the variables in the situation for which you are trying to calculate the force exerted by the magnetic solenoid. For instance, consider a 1 Coulomb (C) charge traveling at 100 meters per second (m/s) through the magnetic field of a solenoid with 1000 turns and 2 amperes (A) of current running through it.

Plug the numbers from your example into the equation and use your calculator to determine the force acting on the charge.
Force = 1 C x 100 m/s x (4 x pi x 10^7 x 1000 x 2 A) = 0.2512 Newtons
The solenoidal electromagnet would exert a force of 0.2512 Newtons on that charge.
Tips & Warnings
 If the charge is traveling at anything other than a 90 degree angle to the magnetic field, the whole force equation should be multiplied by the sine of that angle.
References
 Photo Credit magnetic tape image by Marek Kosmal from Fotolia.com