Magnetic Force on Moving Charges — The Complete Physics Guide
Electric fields push charges in straight lines. Magnetic fields do something stranger: they push charges sideways, perpendicular to whichever way the charge happens to be moving. The result, for a charge moving through a uniform field, is motion along a perfect circle — and finding the speed that makes that circle pass through a specific point is exactly what this game asks you to do.
This isn't just a puzzle mechanic — it's the literal operating principle behind mass spectrometers, particle accelerators, and how physicists identify unknown particles from nothing but the curve of their tracks in a detector.
Why the Force Is Always Perpendicular
The magnetic force on a moving charge is given by F = qv × B — a vector cross product, which by definition always produces a result perpendicular to both of its inputs. That means the magnetic force on a charge is always perpendicular to its velocity, no matter which way the charge is moving.
A force that's always perpendicular to velocity can never speed an object up or slow it down — it can only change the direction of motion. This is exactly what produces circular motion: a constant-magnitude force, always pointing toward a fixed center, is the textbook condition for uniform circular motion.
Deriving the Radius Formula
Setting the magnetic force equal to the centripetal force required for circular motion gives qvB = mv²/r. Solving for r gives the radius formula this game is built around: r = mv / (|q|B). Notice that a faster particle (larger v) traces a larger circle, not a tighter one — intuitive once you see it, but often guessed backwards at first.
Because r depends on mass, this formula is also how a mass spectrometer identifies unknown molecules: ionize them, send them through a known magnetic field at a known speed, and measure how much their path curves. Heavier ions curve less (larger r); lighter ions curve more.
Worked Example — Finding the Required Speed
Problem: A positive charge enters a 2 T field moving in the +x direction. A gate sits at (10, 10) in the same units as the field. What launch speed sends the particle exactly through the gate?
The circle's center lies on the y-axis at (0, r) for a positive charge curving this way. The gate must be exactly distance r from that center: 10² + (10 − r)² = r²
Expanding: 100 + 100 − 20r + r² = r² → 200 = 20r → r = 10
Since r = v/B (with mass and charge magnitude normalized to 1 in this game), v = r × B = 10 × 2 = 20 m/s — exactly the answer to the first level.
Real-World Applications
Mass spectrometry: Chemists and forensic scientists identify unknown compounds by ionizing a sample and measuring exactly how much a magnetic field curves its path — the same r = mv/(|q|B) relationship, solved for mass instead of speed.
Particle accelerators: The Large Hadron Collider and similar machines use powerful magnetic fields to bend high-speed protons into a circular path, keeping them confined to the accelerator ring instead of flying off in a straight line.
Medical imaging: Cyclotrons used to produce medical isotopes for PET scans accelerate charged particles in a magnetic field, spiralling them outward lap after lap — a direct extension of the single-loop physics in this game.
Frequently Asked Questions
What is the formula for the radius of circular motion in a magnetic field?+−
r = mv / (|q|B), where m is mass, v is speed, q is charge, and B is the magnetic field strength. A faster particle traces a larger circle; a stronger field or larger charge produces a tighter one.
Why does a magnetic field make a charge move in a circle?+−
The magnetic force on a moving charge is always perpendicular to its velocity. A force that’s always perpendicular to motion can’t change speed, only direction — which is exactly the condition for uniform circular motion.
Why do positive and negative charges curve in opposite directions?+−
The force direction comes from F = qv×B, which depends on the sign of q. Flipping the charge’s sign flips the force’s direction, so the particle curves the opposite way around even in the exact same field.
Does a magnetic field do work on a moving charge?+−
No — because the force is always perpendicular to velocity, it never has a component along the direction of motion, so it can never speed up or slow down the charge. Its speed stays exactly constant.
How is this used in real mass spectrometers?+−
A mass spectrometer ionizes a sample and fires it through a known magnetic field at a known speed, then measures the radius of the resulting curved path. Since r = mv/(|q|B), measuring r reveals the particle’s mass.