Electromagnetism · force and energy

Charged Particle

Launch into uniform electric and magnetic fields. Replay the recorded path, reverse the charge, and compare kinetic energy with electric work in the same flight record.

Recorded speed $\text{—}$

Instantaneous magnetic radius $\text{—}$

Cyclotron angular frequency $\text{—}$

Kinetic energy $\text{—}$

Electric work $\text{—}$

Energy balance residual $\text{—}$

One flight. One frozen record.

Launch into uniform fields

Recorded pathDraw only samples reached by the replay clock.Velocity and forceArrow lengths are compressed for display.Uniform field directionsMagnetic arrows point normal to the flight plane.Charge

q

1\,\mathrm{C}

Mass

m

1\,\mathrm{kg}

Normal magnetic field

B_z

2\,\mathrm{T}

Horizontal electric field

E_x

0\,\mathrm{V\,m^{-1}}

Vertical electric field

E_y

0\,\mathrm{V\,m^{-1}}

Launch speed

v_0

14\,\mathrm{m\,s^{-1}}

Launch angle from the horizontal

\alpha

0^\circ

Recorded duration

T

6\,\mathrm{s}

Sliders configure the next launch. Replay and scrubbing preserve the record. SI units, nonrelativistic test particle, uniform infinite fields; no radiation, collisions or back reaction. The scene fits the entire flight without resetting an escaping particle.

Nonrelativistic teaching particle in SI units. Uniform prescribed fields; no collisions, radiation or back reaction. Positive horizontal and vertical coordinates form a right-handed plane with the normal magnetic field. Camera scale adapts to the complete recorded flight.

Recorded time

0.00\,\mathrm{s}

Both plots use the frozen flight record. Electric work is calculated from the displacement, independently of the velocity update. A residual reports numerical error, including the position integration error.

\Delta K=W_E,\qquad W_E=q\,\mathbf E\cdot(\mathbf r-\mathbf r_0)

The border is a display frame, not an electrode or a chamber wall. Position scales adapt to the full recorded path; particle size and vector lengths are exaggerated. Positive normal magnetic field follows the right-handed convention.

Explore the potential landscape →MIT: electrostatics and potentials Boris integration and conservation

Start with the essentials

Focus question: Why can a pure magnetic field bend a path into a circle without making the particle faster?
One-sentence intuition: Magnetic force stays perpendicular to velocity and does no work; the electric field directly changes kinetic energy.

Core mathematical model

Work-energy check

K(t)-K(0)=q\,\mathbf E\cdot[\mathbf r(t)-\mathbf r(0)]

Uniform-field electric work comes from displacement. Compare it with the kinetic change computed from recorded velocity. The magnetic field does no work.

Lorentz force

\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)

The cross product sets magnetic magnitude and direction; charge sign reverses the total force.

Gyration and drift scales

r_c=\frac{mv_\perp}{|q|B},\qquad\mathbf v_D=\frac{\mathbf E\times\mathbf B}{B^2}

Mass and perpendicular speed enlarge the orbit; $\mathbf E\times\mathbf B$ drift is independent of mass and charge sign for nonzero charge and magnetic field.

Common difficulties

A magnetic field does not increase speed

Typical misconceptionAny sustained force must make the particle faster.

Better mental modelA perpendicular force rotates velocity; power $q(\mathbf v\times\mathbf B)\cdot\mathbf v$ is always zero.

Negative charge reverses the right-hand result

Typical misconception $\mathbf v\times\mathbf B$ is the force direction for every particle.

Better mental model $\mathbf v\times\mathbf B$ gives the direction for positive charge; $q<0$ reverses it.

Run the experiment

01
Establish pure gyration
Choose Pure magnetic circle and scrub the recorded time.
What to observe: Speed stays fixed while the path becomes circular.
02
Reverse one sign
Reverse charge and launch; compare the recorded path.
What to observe: Rotation reverses while radius stays similar.
03
Add a perpendicular electric field
Choose Crossed-field drift, edit the electric field, then launch a new flight.
What to observe: The orbit center develops steady sideways motion.

Charged Particle

Account for the energy