Problem Statement
Solve the optics problem: A quartz crystal of thickness $t = 1.0$ mm rotates the plane of polarization by $\phi = 21.7°$ at $\lambda = 589$ nm. Calculate the specific rotation $[\alpha]$. $$[\alpha] = \frac{\phi}{t} = \frac{21.7°}{1.0\text{ mm}} = \boxed{21.7°/\text{mm} = 217°/\text{cm}}$$
Given Information
- $[\alpha] = \frac{\phi}{t} = \frac{21.7°}{1.0\text{ mm}$
Physical Concepts & Formulas
Rotational kinematics mirrors linear kinematics with $\theta \leftrightarrow x$, $\omega \leftrightarrow v$, $\alpha \leftrightarrow a$. The angular velocity vector $\boldsymbol{\omega}$ points along the rotation axis (right-hand rule). For a point at distance $r$ from the axis: $v = r\omega$ and $a_\tau = r\alpha$, $a_n = r\omega^2 = v^2/r$.
- $v = r\omega$ — tangential speed from angular velocity
- $a_\tau = r\alpha$ — tangential acceleration
- $a_n = r\omega^2 = v^2/r$ — centripetal acceleration
- $\omega = d\theta/dt$, $\alpha = d\omega/dt$
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$[\alpha] = \frac{\phi}{t} = \frac{21.7°}{1.0\text{ mm}} = \boxed{21.7°/\text{mm} = 217°/\text{cm}}$$
$$\frac{1}{v} = \frac{1}{f} + \frac{1}{u}\quad\text{(mirror)} \quad\text{or}\quad \frac{1}{v} = \frac{1}{f} + \frac{1}{u}\quad\text{(lens with Cartesian)}$$
$$\boxed{\frac{1}{v} – \frac{1}{u} = \frac{1}{f}}$$
Answer
$$[\alpha] = \frac{\phi}{t} = \frac{21.7°}{1.0\text{ mm}} = \boxed{21.7°/\text{mm} = 217°/\text{cm}}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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