Gyro Radius Closes Force Balance - Condensed Matter and Plasma Physics Step by Step

The gyro radius r_L = m * v_perp / (q * B) closes the loop by setting the magnetic force equal to the centripetal force needed for circular motion. That balance is exact only for stated charge, mass, perpendicular velocity, and field, with a uniform B. Real plasmas add field gradients, collisions, pitch-angle spread, relativistic corrections, and measurement uncertainty.

A gyro radius is accepted only when magnetic and centripetal forces match. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Centripetal force closes the orbit

Mass 8 kilograms, speed 3 meters per second, and radius 6 meters give centripetal force 12 newtons.

\frac{mv^2}{r}=\frac{8\cdot 3^2}{6}=12\ \text{N}

The same force is the magnetic force

The magnetic force is also 12 newtons, so the radius closes the force balance.

12\ \text{N}=12\ \text{N}