⭕Bounded Magnetic Field - Find the Centre, Dynamic Circles, Focusing

How to use Charged Particles in a Bounded Magnetic Field

1. 1Choose the field boundary (straight edge, strip, rectangle, circle, triangle) and set the charge sign, field direction, q/m, B and speed
2. 2Move the entry point and launch angle; read the path, central angle and t = θ/(2π)·T, and step through the centre construction
3. 3Open rotating, scaling or shifted circles to see a family of paths, the shaded reachable region and the critical speeds found automatically
4. 4Try random problems on the Practice tab and view the solution with its diagram

FAQ

How do I find the centre of the circular path?

The magnetic force is always perpendicular to the velocity and points at the centre, so the centre lies on the line through the entry point perpendicular to the entry velocity. You need one more line: if you know the exit velocity, draw the perpendicular at the exit point; if you only know the exit point, draw the perpendicular bisector of the chord. Where the lines cross is the centre. Geometry then gives the radius, and r = mv/(qB) gives the speed.

How do I work out the time spent in the field?

Find the central angle θ of the arc, which equals the angle the velocity turns through. Then t = θ/(2π)·T with T = 2πm/(qB), which does not depend on speed. So for one kind of particle, comparing times only means comparing central angles. You can also use t = arc length / v.

What are rotating, scaling and shifted circles?

Rotating circles: same speed, different directions, so every path has the same radius and the centres lie on a circle of radius r around the source; use them to find where particles can land. Scaling circles: same direction, different speeds, so the centres lie on one line perpendicular to the velocity; use them to find the speed at which a path just touches an edge. Shifted circles: identical velocities entering at different points, giving congruent circles.

What do radial entry and magnetic focusing mean in a circular field?

If a particle is aimed at the centre of a circular field, the entry and exit points are symmetric about the line joining the two centres, so it also leaves along a radius, with deflection tan(θ/2) = R/r. When the orbit radius r equals the field radius R, a parallel beam converges on a single point of the circle (focusing), and particles fired in all directions from that point leave parallel (spreading).

What is the critical condition for crossing a strip of field?

At the critical case the path just touches the far edge. For entry perpendicular to the edge the critical radius equals the strip width d, so v = qBd/m. For oblique entry it depends on which way the particle curves: r = d/(1 − sinφ) or d/(1 + sinφ), where φ is the angle from the normal. The scaling-circle tab finds the critical radius and speed by bisection.