The perpendicular force ( FB ) acting upon a positively charged particle moving with a constant velocity ( v ) gives rise to interesting and practical real-world applications:
As the equation shows, the magnitude of this force is greatest when it travels through a magnetic field ( B ) at a 90o angle. As a consequence, the force acting upon the charge in motion is a centripetal or “ center seeking “ force ( Fc ):
Fc = mv2 / r
If the charge in question continuously moves through a magnetic field for a long enough duration, the path it travels will be circular:
Please recall that an acceleration can be linear, but an object moving with a constant velocity that changes direction is accelerating as well. There are, of course, limits to how long such a particle will move in a circular arc without a continuous input of energy. Unto its own ambition, a charged species under the aforementioned conditions will radiate energy and spiral inward towards the center that it seeks to find. The largest ring-shaped particle accelerators have diameters that are miles long, and superconducting electromagnets are fixed around their perimeter. The clever usage of electric fields is also employed in the task of accelerating protons into such a ring with the appropriate energy characteristics. The length of the radius ( r ) of such a system is, in part, determined via the following equivalency:
Fc = FB
qvB sin θ = mv2 / r
sin 90o = 1
qvB = mv2 / r
qB = mv / r
r = mv / qB
Since momentum ( p = mv ) is directly proportional to kinetic energy ( KE ), proportional adjustments between ( KE ) and B-field strength are made until the desired energy is contained in the protons in orbit.
RENAISSANCE PHYSICS 