Physics HL · Chapter 18: Electric and Magnetic Fields
18.4 Crossed Electric and Magnetic Fields (Bridge to Motion-in-Fields)
Use force competition between electric and magnetic terms to derive velocity-selection logic used in beam instrumentation.
Estimated time: 24 minutes
Competing Forces in Crossed Fields
When a charged particle enters a region with both electric and magnetic fields, the net force is the vector sum of electric and magnetic components. If geometry is arranged so these components oppose each other, there can be one speed at which they cancel exactly and the trajectory stays straight.
F_E = qE,qquad F_B = qvB ( ext{for } ec{v}perp ec{B})
Direction is crucial: cancellation requires opposite directions, not just equal magnitudes.
Velocity Selector Condition
qE = qvBRightarrow v = rac{E}{B}
Only particles with this speed pass undeflected in the selected geometry.
Notice that charge magnitude cancels, so selected speed depends on field ratio E/B rather than q. Charge sign still matters for direction of each force; opposite-sign particles deflect opposite ways under the same field arrangement.
Operational Beam-Control Interpretation
This force-balance idea is the conceptual doorway to Chapter 19 (motion in electric and magnetic fields). Here the purpose is to stabilize your force bookkeeping so the later circular and helical trajectory mathematics feels like a direct extension, not a topic reset.
Simulation: Crossed-Field Velocity Selector
Tune E, B, speed, and charge sign to watch when electric and magnetic forces cancel and when the beam deflects.
Balance speed E/B
1.20e+5 m/s
Electric force
3.84e-6 N
Magnetic force
-3.84e-6 N
Predicted path
Straight
Test Yourself
In crossed fields E = 600 N/C and B = 2.0 x 10^-3 T with opposing force directions, enter the undeflected speed.
Hint: Use v = E/B.