Determine the ion's velocity. Though a problem may provide the velocity, typically it must be calculated. In virtually all examples, the ion is accelerated through an electric field, or a "potential difference," given in volts (V) or kilovolts (kV). Multiply the ion's charge with this difference to obtain its energy. Multiply its energy by 2, and divide it by its mass. Take the square root of this quantity to find the ion's velocity. In the equations to determine velocity, the carat, "^," represents exponentiation, and "sqrt" is the square root operation. "V" is the potential difference or voltage, "q" is the ion's charge, "E" is the ion's energy, "m" is the ion's mass and "v" is the ion's velocity. The equations are:
E = charge x potential = q x V
E = 1/2 x mass x velocity^2 = 1/2 m x v^2
v = sqrt(q x E x 2 / m)
Find the magnetic field's force on the ion. Typically, the ion's movement is perpendicular to the magnetic field. If it is not perpendicular in your problem, though, your ion will not have a circular path; it likely will have a helical path. If the movement is perpendicular, multiply the ion's charge, velocity (from Step 1) and the magnetic field strength to obtain the magnetic field's force. With "F" representing the force exerted on the ion and "B" the strength of the magnetic field, the equation is: F = charge x velocity x magnetic field = q x v x B.
Determine the radius of the ion's path. The constant force applied to the particle is perpendicular to its motion, and so its motion is circular. The radius of this circle is determined by the ion's mass and velocity and the force applied to it. Use the quantities you determined in previous steps to find the radius of the ion's path. With "r" representing the radius of the ion's circular path, the equations are:
F = mass x velocity^2 / radius = m x v^2 / r
r = m x v^2 / F
Rearrange the governing equations to reveal a shortcut for this problem. In this case, you don't have to calculate the force at all, though you still must find the ion's velocity. The equations are:
F = q x v x B
r = m x v^2 / F
r = m x v / q x B