= Faraday effect
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= Faraday rotation
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The <Faraday effect> is <magnetic field>-induced <circular birefringence>: opposite <circular polarizations> acquire different <wave phases>, rotating transmitted <linear polarization>. For a transparent medium with <isotropy> before applying the <magnetic field>, in the linear magnetic-response regime at fixed <vacuum> <wavelength> $\lambda_0$ and <temperature> $T$,
$$
\theta=V(\lambda_0,T)\int_0^L B_\parallel(\ell)\,d\ell.
$$
Here $V$ is the <Verdet constant>, $L$ is the material path length, and $B_\parallel$ is the applied <magnetic flux density> component along the ray. This $\mathbf B$ is the external bias, distinct from the oscillating <magnetic field> of the <light> itself. <Michael Faraday> discovered the effect in 1845.<RP Photonics Encyclopedia by Rüdiger Paschotta/Faraday effect>[RP Photonics Encyclopedia, Faraday Effect, Rüdiger Paschotta: What is the Faraday Effect?, rotation law and discovery]{ref}
Fix the sign before using the formula. On the forward path choose a right-handed <Cartesian coordinate system>, propagation along $+z$, and measure $\theta$ from $+x$ toward $+y$ for the <electric field>'s oscillation line. Positive $\theta$ is counterclockwise in the $x$-right, $y$-up end view seen from $z>0$, looking toward the source. Then $B_\parallel=B_z$ and positive $V$ means positive rotation for positive $B_z$. This convention differs from the clockwise-positive convention in <optical rotation>. A <linear polarization> line is unchanged by $180^\circ$; a measured orientation determines accumulated rotation only modulo $\pi$, unless continuity selects a branch.
For uniform $B_\parallel$, the formula reduces to $\theta=VBL$. Its appeal is that a weak change of <refractive index> becomes a visible change of orientation. <Magnetic reversal measurement of Faraday rotation> turns this into an inference of $V$ or of the <magnetic field> path integral. <Faraday rotation on a round trip> explains the directional behavior used by an <optical isolator>. In an astronomical cold <plasma>, the high-<frequency> limit instead leads to the <rotation measure> law $\theta=\mathrm{RM}\lambda_0^2$; that <wavelength> dependence is not a universal law for transparent solids.
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