From voltage to motion

Is a field line a particle trajectory?

Will a particle with sideways velocity exactly follow a field line?

  1. Set up a dipole and show potential.
  2. Release a particle and compare its trajectory with the lines.
  3. Change initial velocity and compare paths in the same field.

What to measure

V in volts, U=qV in joules, particle position and speed.

The idea underneath

A field line represents local direction. A particle carries momentum: acceleration follows qE/m, but velocity need not align with E. For a positive charge descending in potential, potential energy becomes kinetic energy. This distinction matters in electron optics.

U=qV; ΔK=−qΔV; m d²r/dt²=qE

U and K in J; q in C; V in volts; m in kg.

Build the argument

  1. Evaluate each source separately and add its vector contribution. Potential adds as a signed scalar.
  2. Use F = qE or F = q(v × B), then divide by mass to obtain acceleration.
  3. The electric model uses consistent softened potential and field with ε = 0.2 m, so values very near a source differ from ideal point charges.

Worked example 1

A +2 nC charge falling through 10 V gains 20 nJ of kinetic energy.

Worked example 2

For m=10⁻⁹ kg, starting from rest gives v=√(2ΔK/m)=6.32 m/s.

Applications

Electrostatic lenses steer electrons; magnetic fields separate particles by momentum and charge. Both depend on vector direction as much as field strength.

Common misconception

A field line is not generally a particle path. A negative charge accelerates opposite E. A purely magnetic force does no work.

A further investigation

Why can a negative charge gain kinetic energy while moving to higher voltage?

Check your understanding

What does a field arrow directly determine?

  • Positive-charge acceleration direction
  • Every possible initial velocity
Explanation

Force determines acceleration. With q<0 and ΔV>0, ΔK=−qΔV is positive.

Model limits

Static sources and Coulomb superposition. A softened core regularises singularities. Field lines are visual constructions whose density depends on seeding. This is not a time-dependent Maxwell solver.

University Physics Volume 2, 7.2 Electric Potential and Potential Difference

University Physics Volume 2, 5.4 Electric Field