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. Explore a virtual apparatus, collect readings and analyse a graph.
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Adjust the apparatus, then take a reading.
Readings are generated by an ideal model. The diagram is schematic and is not to scale.
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.
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.
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.
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.
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.
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.
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.
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. Suggested independent settings: 0.6, 0.8, 1.0, 1.25, 1.5, 1.75, 2.0 m.
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.
Any lateral or vertical misalignment causes light rays to strike the photometer screen at oblique angles rather than normal incidence. By Lambert's cosine law, oblique illumination reduces irradiance by cos θ, introducing severe systematic errors into the inverse-square distance calculations.
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.
Stray ambient illumination from room windows or ceiling lamps adds directly to the measured illuminance, destroying the strict inverse-square relation E ∝ 1/r². Thermal stabilization ensures that filament resistance and color temperature remain perfectly constant throughout all subsequent trials.
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.
Tungsten incandescent filament luminous intensity is extraordinarily sensitive to operating voltage, scaling approximately as I_lum ∝ V^3.5. Maintaining exact electrical power regulation is indispensable for obtaining reproducible luminous efficacy measurements.
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.
Sampling seven widely spaced distances spanning a four-fold distance range (0.5 m to 2.0 m) produces a sixteen-fold range in illuminance, providing an expansive dynamic range for linear regression of E against 1/r² with high statistical power.
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.
Visual null-detection provides an independent optical verification of the luminous intensity via the classic equality I_x / r_x² = I_s / r_s², corroborating electronic luxmeter readings with direct human psychophysical photometric metrology.
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.
Comparing different lamp technologies under identical geometric and photometric bench conditions demonstrates the dramatic evolutionary leap in lighting efficacy from incandescent blackbody radiation (~16 lm/W) to phosphorescent gas discharge (~70 lm/W) and solid-state bandgap electroluminescence (>110 lm/W).
Points are your observations. The fit uses SI x and y values before display scaling.
| # | Photometer Distance r (m) | Illuminance E (lux) | Inverse-Square Distance 1 / r² (m⁻²) | Illuminance E (lux) | Remove |
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