## Decoding Particle Trajectories in Electromagnetic Fields
Imagine a 3D space where both electric and magnetic fields coexist. When a charged particle enters this arena, its trajectory is governed by the Lorentz force, a beautiful interplay of electricity and magnetism. Let's break down three distinct scenarios to understand how these fields dictate the particle's path.
The Straight Line Path
In our first scenario, we want the particle to move in a straight line along the negative y-axis. For a particle to maintain a perfectly straight trajectory, the net force acting on it must be strictly collinear with its velocity vector. Any perpendicular force component would cause the path to curve.
Let's evaluate the case of a proton starting from rest (v=0) with an electric field E=−E0y^ and a magnetic field B=B0y^. The electric force is simply Fe=qE=−eE0y^. This force accelerates the proton along the negative y-axis. As it gains speed, its velocity vector v becomes parallel to the magnetic field B. Because the magnetic force is given by the cross product Fm=q(v×B), and the cross product of parallel vectors is zero, the magnetic force remains zero throughout the motion. The proton travels in a straight line!
The Velocity Selector
Next, we need the particle to move in a straight line with a constant velocity. This implies zero acceleration, meaning the net electromagnetic force must be exactly zero. This is the principle behind a velocity selector.
Consider an electron moving with velocity v=B0E0y^, in an electric field E=−E0x^ and a magnetic field B=B0z^.
The electric force on the negatively charged electron is Fe=−e(−E0x^)=+eE0x^.
The magnetic force is Fm=−e(v×B)=−e(B0E0y^×B0z^)=−eE0x^.
These two forces are equal in magnitude but opposite in direction. They cancel each other out perfectly, leaving a net force of zero. The electron glides through the fields with a constant velocity.
The Helical Journey
Finally, we desire a helical path with its axis along the positive z-direction. A helix is formed when a particle undergoes circular motion in one plane while simultaneously translating perpendicular to that plane. This requires a magnetic field to provide the centripetal force and an independent velocity component or electric field along the axis of the helix.
Let's look at a proton with an initial velocity v=2B0E0x^, an electric field E=E0z^, and a magnetic field B=B0z^.
The magnetic field B=B0z^ acts on the x-component of the velocity, forcing the proton into a circular path in the xy-plane. Simultaneously, the electric field E=E0z^ exerts a constant force Fe=eE0z^, accelerating the proton along the z-axis. This continuous acceleration means the distance between consecutive loops of the helix—the pitch—will constantly increase. The result is a beautiful, stretching helix along the positive z-axis.
