Photometry & Lamp Efficiency lab report

1. Title and aim

Photometry and Lamp Efficiency. Verify the inverse-square law of illumination E = I / r², determine the luminous intensity I of an electric lamp in candelas (cd) using a precision optical bench photometer, calculate total luminous flux Φ = 4π I in lumens, and evaluate the luminous efficacy η = Φ / P in lumens per watt across incandescent, compact fluorescent (CFL), and solid-state LED light sources.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Luminous Intensity of 100.00 cd, differing by 0.001% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Precision Optical Bench Track: Provides a rigid two-meter graduated steel rail with low-friction sliding riders, anti-backlash clamps, and millimeter vernier scales to measure the exact optical separation between source filaments and photometer detector apertures.
• Primary Standard Lamp Housing: Houses a certified tungsten incandescent standard lamp of known luminous intensity (I_s = 25.00 cd) operated at constant voltage, providing a stable photometric reference baseline.
• Unknown Test Lamp Enclosure: A ventilated light-tight enclosure mounting interchangeable test light sources (75 W incandescent, 14 W CFL, 9 W solid-state LED) with internal baffle vanes preventing stray room reflections.
• Lummer-Brodhun Contrast Photometer Head: An optical comparison assembly utilizing a magnesium carbonate diffuse matte screen and a precision total internal reflection prism cube that presents adjacent concentric viewing fields for visual contrast null balancing.
• Digital Cosine-Corrected Luxmeter Sensor: A silicon photodiode detector equipped with a custom spectral correction filter matched to the CIE V(λ) photopic human eye response curve and a diffuse opal glass diffuser ensuring Lambertian cosine angular response.
• Digital Power Analyzer and Regulated Power Supply: Supplies stable low-distortion AC/DC electrical excitation while simultaneously recording true RMS voltage V, circuit current I, power factor cos φ, and active electrical power P with 0.1% accuracy.
• Nominal Test Lamp Operating Power: P_elec = 75.00 W (Incandescent Tungsten Reference Source)
• Nominal Standard Reference Luminous Intensity: I_standard = 25.00 cd at certified operating potential
• Detector Spectral Sensitivity Filter: Matched to CIE 1931 Standard Photopic Eye Response V(λ)
• Ambient Environment: Darkened optical enclosure with matte black baffles eliminating stray scatter

4. Theory and graph choice

Photometry is the branch of optical physics that quantifies radiant energy evaluated according to the visual sensitivity of the human visual system. In 1760, Johann Heinrich Lambert formulated the foundational laws of photometry, demonstrating that illumination varies inversely with the square of the distance from a point source and directly with the cosine of the angle of incidence. In 1843, Robert Bunsen developed the grease-spot photometer, later perfected in 1889 by Otto Lummer and Eugen Brodhun with their precision contrast prism cube, establishing optical bench metrology as the primary standard for comparing unknown light sources against certified incandescent standards. In this experiment, students explore the fundamental inverse-square law of illumination E = I / r² using a precision two-meter graduated optical bench. By systematically altering the distance r between a test lamp and a cosine-corrected digital luxmeter detector, students measure illuminance E across distances from 0.50 m to 2.00 m. A linear regression of E against 1/r² extracts the source's luminous intensity I in candelas (cd), isolates any stray ambient background illuminance as the intercept, calculates the total luminous flux Φ = 4π I in lumens, and computes the luminous efficacy η = Φ / P in lumens per watt across incandescent, compact fluorescent (CFL), and modern solid-state LED lamps.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units cd.

5. Method

1. Mount the optical bench on a vibration-isolated laboratory table and adjust the leveling screws until the spirit level indicates zero tilt along both longitudinal and transverse axes. Install the test lamp housing, photometer contrast head, and sensor carrier on their sliding riders. Adjust the vertical and lateral spindle thumbscrews until the centers of the lamp filament, photometer diffusion screen, and detector aperture are collinear with the central optical axis of the track.
2. Close the light-tight shielding curtains around the optical bench to eliminate ambient room lighting. Energize the digital luxmeter and power analyzer without lighting the lamps, and verify that the residual dark-current display registers zero lux (tare zero adjustment). Light the primary reference standard lamp and allow it to operate for 10 minutes until thermal and filament resistance equilibrium are achieved.
3. Mount the test incandescent lamp (75 W nominal rating) into the test socket and connect its terminals to the regulated power analyzer. Adjust the power supply voltage until the operating voltage reaches exactly 120.0 V RMS. Record the active power consumption P (in Watts), current I_A, and verify that the electrical input remains steady within ±0.05 W.
4. Position the photometer detector at an initial separation distance r = 0.60 m from the effective center of the lamp filament. Verify that the detector face is perpendicular to the beam axis. Record the steady-state illuminance E displayed on the calibrated luxmeter. Systematically increment the distance to r = 0.80 m, 1.00 m, 1.25 m, 1.50 m, 1.75 m, and 2.00 m, recording the steady illuminance at each station.
5. Replace the photoelectric sensor with the Lummer-Brodhun contrast photometer head, placing the certified standard lamp (I_s = 25.0 cd) at the opposite end of the 2.00 m track. Slide the photometer carriage along the bench until the concentric circular fields viewed through the eyepiece exhibit identical brightness and zero contrast boundary. Record the balancing distances r_x and r_s.
6. Replace the 75 W incandescent lamp sequentially with a 14 W compact fluorescent lamp (CFL) and a 9 W solid-state LED lamp designed for equivalent illumination. For each source, record the steady-state electrical power P, determine the luminous intensity I at r = 1.00 m, calculate total luminous flux Φ = 4π I, and compute the luminous efficacy η = Φ / P in lumens per watt.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0009%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 0.00708; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 0.01694. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 0.02212.

Propagating only the relevant fit coefficient gives a fit-only standard error of 0.01694 cd. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Photometer detector cosine error at slightly oblique ray incidence (cos θ deviations contributing up to ±0.5% if detector surface is tilted by >1°), optical bench track millimeter scale parallax in distance measurement (±1.0 mm in r, magnified as 2(δr/r) ≈ 0.4% in 1/r² at r = 0.50 m), stray ambient room illumination and specular reflections from bench hardware or nearby walls (mitigated by enclosed light-tight velvet hoods and matte black baffle apertures, with residual stray light captured as the positive vertical intercept b on the regression line), AC mains line voltage fluctuations affecting lamp operating temperature (since tungsten filament luminous intensity scales as I_lum ∝ V^3.5, regulated within ±0.05% using a true RMS AC power conditioner), spectral mismatch between the silicon detector filter and the standard CIE V(λ) photopic eye curve (spectral correction factor f₁′ ≤ 1.5%), and finite lamp filament dimensions deviating from an ideal point source at close distances (<0.08% at r = 0.50 m) govern the uncertainty envelope. Ordinary least-squares regression of illuminance E against inverse-square distance 1/r² averages out random detector quantization and voltage micro-fluctuations, with the slope isolating the pure luminous intensity I with sub-0.01% error and the vertical intercept isolating any baseline ambient background illuminance.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Ensure the optical bench track is completely leveled and aligned so that light strikes the photometer detector at exact normal incidence (θ = 0°).
• Perform all photometric readings inside a darkened room or enclosed light-tight baffle hood to eliminate stray ambient room lighting.
• Allow incandescent and fluorescent lamps at least 10 minutes to reach thermal and electrical equilibrium before recording precision illuminance readings.
• Maintain the test lamp operating voltage strictly constant using a regulated power supply and verify input electrical power with a calibrated true RMS digital power analyzer.
• Never touch quartz-halogen or high-temperature lamp glass envelopes with bare fingers, as skin oils cause localized hot-spots and premature devitrification.

Calculations assume an idealized point source obeying E = I / r² with isotropic spherical emission, negligible absorption or scattering in ambient laboratory air, and zero specular reflection from optical bench track rails. Real incandescent lamps exhibit subtle filament coiling shadows that produce minor directional variations in intensity (typically ±3% between axial and equatorial planes), while finite detector aperture diameter (25 mm) creates a minute second-order geometric deviation (<0.08% at r = 0.50 m) from ideal point-source geometry.

12. Conclusion

The worked example illustrates how to obtain Luminous Intensity from measurements, graph variables and a fitted coefficient. The estimate is 100.00 cd. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.