Physics HL · Chapter 19: Motion in Electric and Magnetic Fields
19.2 Motion in Uniform Magnetic Fields
Use Lorentz-force geometry to classify straight-line, circular, and mixed-component trajectories in magnetic fields.
Estimated time: 44 minutes
Lorentz Force Geometry and Special Cases
A magnetic field acts only on moving charge and the force is perpendicular to both velocity and field. If v is parallel or anti-parallel to B, sin(theta) is zero and there is no magnetic force, so the particle continues straight. The most important nontrivial case is v perpendicular B, where force magnitude is maximal and always perpendicular to velocity.
F_B = |q|vBsin heta,qquad ec{F}_B = q, ec{v} imes ec{B}
Use right-hand rule for positive charge and reverse direction for negative charge.
Direction is not optional bookkeeping here; it determines the curvature side. In page diagrams, dot means out of page and cross means into page. Combine that with velocity direction to determine whether the path bends clockwise or counterclockwise.
Circular Motion from v Perpendicular B
When magnetic force is always perpendicular to velocity, it acts as centripetal force. Setting qvB equal to mv^2/r gives the standard radius relation. This equation is one of the most-used forms in exam questions involving particle beams and isotope separation.
|q|vB = rac{mv^2}{r}Rightarrow r = rac{mv}{|q|B}
Larger speed and mass increase radius; larger charge magnitude or magnetic field decrease radius.
Radius trends are physically intuitive: heavy or fast particles are harder to bend, while stronger fields bend paths more tightly. In mixed-ion beams this is exactly why different species separate spatially in a magnetic analyzer.
Period and Frequency in Magnetic Circular Motion
Substituting the radius relation into T = 2pi r / v removes v entirely. This is a powerful result: for non-relativistic speeds in fixed B, period depends only on m and q, not on speed. Faster particles have larger radius but they also move faster along that larger circle in exactly compensating fashion.
T = rac{2pi m}{|q|B},qquad f = rac{1}{T} = rac{|q|B}{2pi m}
This speed-independence underpins cyclotron frequency targeting at moderate energies.
Important
Magnetic force changes direction of velocity, not speed, in a uniform B-only region.
Simulation: Circular and Helical Magnetic Paths
Vary q, m, B, and launch angle to see when motion is straight, circular, or helical, and track radius/period scaling in real time.
Path class
Circle (v perpendicular B)
Radius
0.217 m
Period
2.62e-7 s
Frequency
3.81e+6 Hz
Helix pitch
8.35e-17 m
Magnetic force magnitude
2.08e-13 N
v perpendicular
5.20e+6 m/s
v parallel
3.18e-10 m/s
Test Yourself
An electron with speed 3.2 x 10^7 m/s enters a 0.025 T field at right angles. Enter the radius of its circular path.
Hint: Use r = mv/(|q|B) with me = 9.11 x 10^-31 kg and e = 1.60 x 10^-19 C.