Photometry

Fizeau's method uses a rotating toothed wheel to intermittently block light that travels to a distant mirror and back. By adjusting the wheel's rotation speed until the reflected light is completely blocked:

• Light passes through a gap, travels to mirror and back
• When it returns, if a tooth now blocks the path, no light is seen
• Speed of light can be calculated using: c = 2d × N × n, where:
  • d = distance to mirror
  • N = number of teeth
  • n = rotational frequency when light disappears

Example: With 180 teeth and 12.0 km distance, the minimum angular speed for light disappearance is 1.25 × 10⁴ deg/s

Foucault's method improved earlier techniques by:

• Allowing laboratory-scale measurements (no long distances required)
• Enabling speed of light measurement in different media (like water)

Key components:

1. Rotating mirror (high angular velocity)
2. Fixed mirror (at distance)
3. Light source and lens system

Working principle:

• Light reflects from rotating mirror to fixed mirror and back
• During travel time, rotating mirror turns slightly
• Returning beam reflects at different angle
• Angular displacement is proportional to light's travel time
• c = 8πLNr/θ where:
  • L = distance between mirrors
  • N = rotation rate
  • r = distance from rotating mirror to observation point
  • θ = observed angular displacement

Difference between the two efficiency measures:

1. Luminous efficiency:

   • Ratio of luminous flux to radiant flux
   • Measures how effectively radiation produces visible light
   • Maximum theoretical value: 683 lm/W at 555 nm
   • Formula: Luminous flux ÷ Radiant flux

2. Overall luminous efficiency:

   • Ratio of luminous flux to power input
   • Measures total system efficiency from energy input to light output
   • Formula: Luminous flux ÷ Power input to source

Limiting factors for overall luminous efficiency:

• Thermal losses (especially in incandescent bulbs)
• Emission in non-visible wavelengths (IR, UV)
• Conversion inefficiencies in phosphors
• Driver/ballast losses in electronic components
• Spectral distribution not matching peak human sensitivity
• Quantum efficiency limitations in LEDs

Typical values:

• Incandescent: 10-15 lm/W
• Fluorescent: 50-100 lm/W
• LED: 70-150 lm/W
• Theoretical maximum: ~300 lm/W (white light)

Working principle of Bunsen photometer:

1. Setup components:

   • Paper screen with oil spot (creates a translucent area)
   • Two light sources positioned on opposite sides
   • Movable screen on optical bench

2. Physical basis:

   • Oil spot appears darker than surroundings when viewed from more illuminated side
   • Oil spot appears brighter than surroundings when viewed from less illuminated side
   • Oil spot disappears when illuminance is equal on both sides

3. Measurement procedure:

   • Position screen between sources and adjust until oil spot "disappears"
   • At equilibrium, illuminance from both sources is equal: E₁ = E₂
   • Apply inverse square law: I₁/d₁² = I₂/d₂²
   • Therefore: I₁/I₂ = d₁²/d₂²

4. Practical considerations:

   • Screen must be perpendicular to line joining sources
   • Background lighting must be eliminated
   • Observer's position should be consistent
   • Multiple readings improve accuracy

This method directly applies the inverse square law to determine the ratio of intensities without requiring absolute measurements.

Relative luminosity and human visual perception:

1. Definition: Relative luminosity measures how efficiently different wavelengths produce brightness sensation relative to the most effective wavelength (555 nm).

2. Characteristics:

   • Peaks at 555 nm (yellow-green) = 1.0
   • Falls to ~0.1 at 450 nm (blue) and 650 nm (red)
   • Nearly zero beyond 400 nm and 700 nm
   • Based on average human photopic (daylight) vision

3. Physiological basis:

   • Corresponds to sensitivity of cone cells in retina
   • Evolution optimized human vision for sunlight spectrum
   • Different distribution for scotopic (night) vision (peaks ~507 nm)

4. Practical significance:

   • Explains why equal-energy yellow light appears brighter than red or blue
   • Forms basis for photometric measurements (lumens, candela)
   • Critical for lighting design and efficiency calculations
   • Explains why green laser pointers appear brighter than red ones of equal power

5. Mathematical representation: V(λ) function (CIE standard luminosity function)

• Relative luminosity forms a bell-shaped curve across the visible spectrum (400-700 nm)
• Values: Near zero at 400 nm (violet) and 700 nm (red), peaks at 1.0 around 555 nm (yellow-green)
• Drops more steeply toward red than toward blue wavelengths
• Sensitivity at 500 nm (blue-green) ≈ 0.7
• Sensitivity at 600 nm (orange) ≈ 0.6

Practical implications:

• Energy-efficient lighting should focus on wavelengths close to 555 nm for maximum brightness per watt
• Red lighting requires more energy than green lighting for equivalent perceived brightness
• Blue lighting appears dimmer than green lighting of equal power
• LED displays must compensate with higher intensity blue and red components to achieve color balance
• Warning signs most effective with yellow-green coloration for maximum visibility

• Relative luminosity: The ratio of brightness-producing capacity of light at a given wavelength compared to light at 555 nm (where it equals 1.0)
• Dimensionless quantity ranging from 0 to 1.0
• Determined by physiological response of human photoreceptors

To calculate luminous flux of a monochromatic source:

1. Identify the wavelength of the source
2. Find the corresponding relative luminosity value
3. Multiply the radiant flux (in watts) by the relative luminosity value
4. Multiply by the conversion factor 685 lumen/watt

Formula: Luminous flux (lumen) = Radiant flux (watt) × Relative luminosity × 685 lumen/watt

Example: For a 10W source at 600 nm with relative luminosity of 0.6: Luminous flux = 10W × 0.6 × 685 lumen/watt = 4,110 lumen

The solid angle subtended by a small area ΔA at a point source is:

$\Delta\omega = \frac{\Delta A \cos\theta}{r^2}$

Where:

• ΔA is the small area
• r is the distance from the source to the area
• θ is the angle between the normal to the area and the line connecting the source to the area

This solid angle determines the luminous flux (ΔF) passing through the area according to:

$\Delta F = I \Delta\omega = I \frac{\Delta A \cos\theta}{r^2}$

Where I is the luminous intensity of the source in that direction.

The cosθ factor accounts for the effective area as seen from the source's perspective (projection of the area perpendicular to the light ray).

Derivation of the illuminance equation from first principles:

1. Start with luminous flux (ΔF) through solid angle (Δω): $\Delta F = I \Delta\omega$ Where I is luminous intensity in candelas

2. The solid angle subtended by area ΔA is: $\Delta\omega = \frac{\Delta A \cos\theta}{r^2}$

3. Substituting this into the flux equation: $\Delta F = I \frac{\Delta A \cos\theta}{r^2}$

4. Illuminance is flux per unit area: $E = \frac{\Delta F}{\Delta A} = \frac{I \cos\theta}{r^2}$

Physical meaning of each term:

• I: Source's inherent brightness (luminous intensity) in a specific direction
• 1/r²: Geometric spreading of light energy over increasingly larger areas
• cosθ: Projection effect - when surface is tilted, same energy spreads over larger area

This equation combines the inverse square law (1/r²) with Lambert's cosine law, representing how light energy disperses through space and interacts with surfaces.

Lambert's Cosine Law states that for a perfectly diffused emitting surface:

• The luminous intensity (I) varies with the angle (θ) from the normal according to: I = I₀cosθ
• Maximum intensity (I₀) occurs along the normal to the surface
• Intensity decreases as the angle from normal increases
• At 90° from normal, intensity becomes zero

This law explains why flat light sources appear equally bright from different viewing angles despite radiating different amounts of light in different directions.

Example: A flat LED panel that appears equally bright whether viewed straight-on or at an angle, despite emitting more total light in the forward direction.