17-Phys-A7 Optics · December 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
17-Phys-A7, Optics — National Exams, December 2018. 3 hours; closed book (approved Sharp/Casio calculator only). Total 78 marks. Questions 1–6 are mandatory; the paper then offers a choice of Question 7 or 8, and a choice of Question 9 or 10. Every question is solved in full below as a complete study resource, including both members of each either/or pair.
Reference texts. Hecht, Optics, 5th ed.; Pedrotti, Pedrotti & Pedrotti, Introduction to Optics, 3rd ed.; Griffiths, Introduction to Electrodynamics, 4th ed. (Ch. 9, EM waves in matter, for Maxwell's equations).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Part (a) — first-order diffraction angle of the red ($656$ nm) line.
Given. Grating groove density $1000$ lines/cm, red line $\lambda=0.656\ \mu\text{m}=656$ nm, normal incidence, order $m=1$.
Find. Diffraction angle $\theta_1$.
Approach. Grating equation $d\sin\theta_m=m\lambda$, with groove spacing $d=1/(\text{lines per unit length})$.
| Quantity | Value |
|---|---|
| Groove spacing $d$ | $10\ \mu$m |
| 1st-order angle, red line ($656$ nm) | $3.76^\circ$ |
Part (b) — design of a grating spectrometer. A minimal, accurate instrument is the classic Czerny–Turner grating monochromator/spectrometer. Light from the unknown source enters through a narrow entrance slit at the focus of a collimating concave mirror, which sends a parallel beam onto the plane reflection grating (mounted on a precision rotation stage with an angular vernier or digital encoder reading $\theta_i$, the grating's rotation angle). The grating diffracts each wavelength present in the source to its own angle $\theta_m(\lambda)$ via $d\sin\theta_i+d\sin\theta_m=m\lambda$ (the general grating equation, allowing non-normal incidence); a second concave mirror focuses each diffracted beam to a point on the focal (exit) plane, where either a scanning exit slit + single detector, or a linear CCD/photodiode array, records intensity vs. position.
Measurement procedure. To measure an unknown wavelength, either (i) scan the grating angle with the detector fixed behind an exit slit and record the drive angle at each intensity peak, or (ii) use the array detector and convert each illuminated pixel's position $x$ on the focal plane to a diffraction angle via the focusing mirror's focal length $f_2$ ($\theta_m=\arctan(x/f_2)$ relative to the mirror axis), then invert the grating equation for $\lambda$ at that $(\theta_i,\theta_m,m)$.
Calibration. Before measuring the unknown source, the instrument is calibrated against a reference source of precisely known emission lines (e.g. a mercury or neon calibration lamp, or the hydrogen lines themselves once independently verified) — recording the grating angle (or pixel position) at each known reference wavelength establishes the angle-to-wavelength (or pixel-to-wavelength) mapping and removes any fixed mechanical offset in the angle encoder or slit alignment.
Accuracy and uncertainty. Differentiating the grating equation at fixed $\theta_i$, $d\cos\theta_m\,d\theta_m=m\,d\lambda$, so the wavelength uncertainty from an angular reading uncertainty $\delta\theta_m$ (set by the encoder resolution, or by pixel size $\delta x$ via $\delta\theta_m\approx\delta x/f_2$) is $\delta\lambda\approx\dfrac{d\cos\theta_m}{m}\,\delta\theta_m$ — finer angular resolution, a higher groove density $1/d$, and a longer focal length $f_2$ (which reduces $\delta\theta_m$ for a given detector pixel size) all directly reduce the wavelength uncertainty. Slit width sets a competing limit: a wider entrance slit passes more light (better signal-to-noise) but blurs each spectral line over a larger angular range, degrading resolution — the standard resolving-power figure of merit is $R=\lambda/\delta\lambda=mN$, where $N$ is the total number of illuminated grooves, so a wider illuminated grating aperture (more grooves intercepted) directly improves the achievable resolution independent of slit width, up to the point the slit itself becomes the limiting factor.
