= Solution
The <Faraday effect> rotates the plane of <linear polarization> when light propagates through a material with a <magnetic field> component along the ray. In the weak-field transparent regime,
$$
\boxed{\theta=V(\lambda,T)\int\mathbf B\cdot d\mathbf l},
$$
with a stated propagation and transverse-basis convention; $V$ is the <Verdet constant>. For a uniform longitudinal field and sample length $L$, the magnitude is $|VBL|$. A scalar intensity change without a corresponding polarization analysis would not establish this effect.
The microscopic field breaks the equivalence of the two <circular polarization> modes: the <Lorentz force> and the Zeeman shifts of optical transitions produce different susceptibilities for their opposite senses of rotation. A linearly polarized wave is an equal-amplitude superposition of these two modes. If their real refractive <refractive indices> are $n_+,n_-$, their relative <wave phase> after the sample is $\Delta\phi=(\omega L/c)(n_+-n_-)$. Recombination rotates the line by
$$
\boxed{\theta=\frac{\Delta\phi}{2}}.
$$
Equal transmitted circular <amplitudes> preserve <linear polarization>. Unequal <absorption> produces <circular dichroism> and generally <elliptical polarization> as well, so pure rotation is an approximation appropriate away from strong <absorption>.
A modern motivating experiment sends polarized <laser> light through a <glass> rod inside a <solenoid>, then through a rotatable analyzer onto a <photodiode>. Scan the analyzer first to calibrate its angular response. Driving the <solenoid> sinusoidally and recording the detector synchronously separates a small rotation signal from unsynchronized noise; the conversion to angle still requires the analyzer slope and the magnetic-field integral through the rod. A first-party William & Mary laboratory manual specifies this apparatus and measurement procedure.<Physics 251: Experimental Atomic Physics, Fall 2019>[Faraday Rotation laboratory chapter, pp. 2–7: solenoid, glass rod, analyzer calibration and synchronized readout]{ref}
Additional controls are equal-magnitude field reversals, zero-field readings before and after each pair, and a blocked-beam pickup test. The odd-in-field rotation is $[\theta_{\rm obs}(B)-\theta_{\rm obs}(-B)]/2$; a fixed analyzer offset cancels. Heating, stressed-glass <birefringence>, source drift and field-dependent transmission need separate checks. A single reading near crossed extinction loses the sign because its leading response is quadratic; an analyzer near $45^\circ$ gives a linear small-rotation response.
<Faraday rotation> is nonreciprocal. With the magnetic bias held fixed, retracing the path adds rotation in the same laboratory sense rather than undoing it, unlike reciprocal natural <optical activity>. This permits an <optical isolator> with a $45^\circ$ Faraday rotator between appropriately oriented <polarizers>. One must compare the returned polarization in a consistently defined laboratory basis; merely reversing the sign of $\mathbf B\cdot\widehat{\mathbf k}$ while retaining the old ray-basis convention gives an incorrect cancellation.
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