= Electron optics
{wiki}
<Electron optics> studies how <electric fields> and <magnetic fields> guide <Electrons> and form images. The <Lorentz force> supplies a trajectory description, while the <de Broglie wavelength> supplies the <wave> scale needed for <diffraction>. Both descriptions matter: steering a narrow beam and resolving two nearby specimen features are different achievements.
For a <magnetic electron lens>,
$$
\mathbf F=-e\mathbf v\times\mathbf B,\qquad \mathbf F\cdot\mathbf v=0.
$$
Here $e>0$ is the <elementary charge>, $\mathbf v$ the <Electron> <velocity>, and $\mathbf B$ the <magnetic field>. The second equality follows because a <cross product> is perpendicular to each factor: a static <magnetic field> bends the trajectory without increasing the <kinetic energy>. Acceleration instead requires work by an <electric field>. This distinction lets an <electron microscope> adjust imaging with <magnetic fields> while an accelerating <voltage> sets the <Electron> <momentum>.
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