EMG-08

Lorentz force

F = q (E + v B) for perpendicular v and B.

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Charged particlesLorentz 1892Heaviside

Governing equation

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

$v \in \left[0, 1\times 10^{6}\right]$

where

q: Charge (µC)
E: Electric field (N/C)
v: Speed (m/s)
B: Magnetic field (mT)
F: Force magnitude (N)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

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

$v \in \left[0, 1\times 10^{6}\right]$

$v = 0$ $F = 3\times 10^{-4}$

Along the curve

②

Domain of the variable — blue

v_{\min} = 0,\quad v_{\max} = 1\times 10^{6}

$v \in \left[0, 1\times 10^{6}\right]$

$v = 0$ $v = 0$

Along the curve

③

Response curve — green

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

$v \in \left[0, 1\times 10^{6}\right]$

$v = 0$ $F = 3\times 10^{-4}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}F}{\mathrm{d}v}

$v \in \left[0, 1\times 10^{6}\right]$

$v = 6289.3$ $F^{\prime} = 7.5\times 10^{-9}$

Along the curve

Lecture brief

Historical brief

Coulomb, Gauss, Ampère, Faraday and Maxwell (1861–65) unified charge, current and light. The lab computes fields, induction, Poynting flux and the electromagnetic wave in SI. This sheet (EMG-08 — Lorentz force) is the form associated with Lorentz 1892 · Heaviside. Working symbols: $q$ , $E$ , $v$ , $B$ $\rightarrow$ $F$ . The Lorentz force is the operational definition of the fields E and B acting on a point charge.

Purpose

Purpose: compute

F

from

q

E

v

B

in Electromagnetism via

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

F = q (E + v B) for perpendicular v and B. Use it when a real electromagnetism question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

q = 1.500\,\mathrm{\mu C}

E = 200.000\,\mathrm{N/C}

v = 40.000\,\mathrm{m/s}

B = 5.000\,\mathrm{mT}

, the governing relation

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

yields

F = 3.003e-4\,\mathrm{N}

. v perpendicular to B so |v×B| = v B; E parallel to F. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Charge q1.5 µCElectric field E200 N/CSpeed v40 m/sMagnetic field B5 mT

Outputs

Force magnitude F0.0003 N

Reading speed

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Narration of this film

v perpendicular to B so |v×B| = v B; E parallel to F.

The Lorentz force is the operational definition of the fields E and B acting on a point charge.

Reading speed

Watch on YouTube