Historical Context & Motivation
The study of charged particles moving through electromagnetic fields has shaped nearly every branch of modern physics and technology. From the first cathode-ray experiments that revealed the electron to the massive particle accelerators that probe the structure of matter, scientists have relied on the interplay between electric and magnetic forces to steer, accelerate, and identify charged particles. Understanding how charges behave in these fields is not just an abstract exercise — it's the foundation of everything from television screens to cancer-treatment proton beams.
The central question of IB Physics topic D.3 is deceptively simple: What path does a charged particle follow when it enters an electric field, a magnetic field, or both? Answering this requires combining Coulomb's law, Newton's second law, and the magnetic force rule into a unified toolkit. The sections ahead will equip you with that toolkit and show you how to apply it to real IB-style problems.
Core Principles & Definitions
Before diving into calculations, you need a firm grasp of four core ideas that govern how charged particles move through electromagnetic fields. Each principle connects force, acceleration, and trajectory in a specific way. Together, they form the conceptual backbone of every problem you will encounter in D.3.
Electric Force on a Charge
Magnetic Force on a Moving Charge
Circular Motion in B Fields
Parabolic Paths in E Fields
Velocity Selector (Crossed Fields)
Visual Explanation — Paths in E and B Fields
The diagram below compares the two fundamental trajectories side by side. On the left, a positive charge enters a uniform electric field between parallel plates and curves in a parabola — exactly like a ball thrown horizontally under gravity. On the right, the same charge enters a region of uniform magnetic field directed into the page; the perpendicular force bends it into a perfect circle. Study the force arrows carefully: in the electric-field case, force is always downward; in the magnetic-field case, force always points toward the center of the circular arc.
Notice the key difference: in the electric-field diagram, the force arrow (red) always points straight down regardless of where the particle is, just like gravity on a projectile. In the magnetic-field diagram, the force arrow always points radially inward toward the center of the circle. This is why the magnetic force does no work — it is always perpendicular to displacement. The particle's kinetic energy stays constant even though its direction keeps changing.
Mathematical Framework
Three core equations form the mathematical backbone of D.3. Each one connects force, motion, and field quantities in a different scenario. You should be able to recall these from memory and identify when each applies.
Detailed Breakdown — Key Scenarios
IB D.3 problems typically fall into a handful of recognizable scenarios. Knowing which scenario you are dealing with is half the battle. The table below summarizes the most common setups, the shape of the resulting path, and the key equation you need.
| Scenario | Field(s) Present | Path Shape | Key Equation |
|---|---|---|---|
| Charge at rest in E | Uniform E only | Straight line (accelerating) | F = qE, then a = F/m |
| Charge enters E ⊥ to v | Uniform E only | Parabola (projectile analogy) | a = qE/m in field direction |
| Charge enters B ⊥ to v | Uniform B only | Circle | r = mv/(qB) |
| Charge enters B at angle θ | Uniform B only | Helix (spiral) | r = mv sin θ/(qB); pitch from v cos θ |
| Crossed E and B (velocity selector) | E ⊥ B ⊥ v | Straight line (if v = E/B) | v = E/B |
The velocity selector diagram illustrates a concept that appears frequently in IB exams. Notice how the selector is mass-independent: any particle with the correct speed passes through, regardless of its mass or charge magnitude. Once the selected particles exit, they often enter a pure magnetic field region where they separate by mass — this is the principle behind the mass spectrometer. The radius of curvature in that second region, r = mv/(qB), depends on mass, so different isotopes land at different positions on a detector.
Worked Example
Let's work through a full IB-style problem that combines several of the ideas we've covered. Pay close attention to the strategy of identifying the scenario, selecting the right equation, and checking units at each step.
Electric vs. Magnetic Fields — A Comparison
Students often confuse the effects of electric and magnetic fields on charged particles. The table below highlights the most important differences. Memorizing these distinctions will help you quickly identify which equations to use and what path to expect.
| Property | Electric Field (E) | Magnetic Field (B) |
|---|---|---|
| Force direction | Parallel (or anti-parallel) to field | Perpendicular to both v and B |
| Force depends on speed? | No — F = qE regardless of v | Yes — F = qvB sin θ |
| Acts on stationary charges? | Yes | No — particle must be moving |
| Work done on charge | Can do positive or negative work (changes KE) | Zero (F ⊥ v always) |
| Typical path | Straight line or parabola | Circle or helix |
| Changes speed? | Yes | No — only direction changes |
Connection to Advanced Theory
The physics of D.3 lays the groundwork for more advanced topics you may encounter in university physics or the IB HL extension. The table below maps key D.3 ideas to their advanced counterparts, showing how the same principles scale up to more complex and powerful models.
| D.3 Concept | Advanced Extension |
|---|---|
| F = qvB (magnetic force) | Lorentz force: F = q(E + v × B), combining both fields in a single vector equation |
| r = mv/(qB) at low speed | Relativistic radius: r = γmv/(qB), where γ is the Lorentz factor — needed when v approaches c |
| Circular motion in uniform B | Cyclotron frequency ω = qB/m, which is independent of radius and speed (non-relativistic limit) |
| Velocity selector (E/B) | Wien filter in mass spectrometry; Hall effect in solid-state physics |
| Helical motion (v at angle to B) | Magnetic mirroring and plasma confinement in fusion reactors (tokamaks) |
One of the most elegant extensions is the idea of cyclotron resonance. In a cyclotron, the time to complete each semicircle is always the same (T = 2πm/(qB)) regardless of the radius, because as the particle speeds up and its radius grows, it also has farther to travel — the two effects exactly cancel. This means you can use a fixed-frequency alternating voltage to accelerate the particle every half-turn. The breakdown of this elegant feature at relativistic speeds motivated the invention of the synchrotron, which adjusts the frequency as the particle gains energy.
Practice Problems
Lesson Summary
Charged particles in electric fields experience a force F = qE that acts parallel to the field, changing both speed and kinetic energy. A charge entering perpendicular to a uniform E field follows a parabolic path, exactly analogous to projectile motion. In contrast, the magnetic force F = qvB sin θ is always perpendicular to velocity, so it does no work and produces uniform circular motion with radius r = mv/(qB) when v is perpendicular to B.
When both fields are present and perpendicular, the velocity selector condition v = E/B allows only particles of one specific speed to pass through undeflected. This principle underpins the mass spectrometer and many particle accelerators. To solve D.3 problems, identify the scenario (E only, B only, or crossed fields), select the correct equation, apply Newton's second law or energy conservation, and always check whether the magnetic force does work (it doesn't). These tools will carry you confidently through any IB exam question on motion in electromagnetic fields.