Photometry

Step 1: Identify the condition for light disappearance

• Light must return to a tooth position after traveling 2d distance

Step 2: Calculate time for light to travel to mirror and back

• t = 2d/c = (2 × 10,000 m)/(3 × 10⁸ m/s) = 6.67 × 10⁻⁵ s

Step 3: Determine rotation needed during this time

• Wheel must rotate 1/(2 × 200) = 1/400 of a full turn
• Angular displacement = (1/400) × 360° = 0.9°

Step 4: Calculate required angular speed

• ω = 0.9°/(6.67 × 10⁻⁵ s) = 13,493 degrees/second ≈ 1.35 × 10⁴ deg/s
• In rpm: (13,493 deg/s)/(360 deg/rev) × 60 s/min = 2,249 rpm

This is the minimum speed (other than zero) at which the reflected image disappears.

A polygonal mirror with more faces in Michelson's method provides several advantages:

1. Increased data collection rate:

   • More reflective surfaces available per revolution
   • Greater number of light pulses per unit time
   • Statistical averaging improves measurement precision

2. Lower rotation speeds needed:

   • For N faces, rotation speed can be reduced by factor of N compared to single mirror
   • Reduces mechanical stress and bearing wear
   • More stable rotation with fewer vibration issues

3. Better signal-to-noise ratio:

   • More frequent sampling
   • Improved averaging of measurements
   • Reduced impact of random fluctuations

4. Practical measurement benefits:

   • Easier to achieve precise angular positioning
   • More uniform reflections
   • Enhanced experimental reproducibility

Relationship between luminous flux and radiant flux:

• Radiant flux: total energy emitted per unit time (measured in watts)
• Luminous flux: measure of the perceived brightness power, weighted by human eye sensitivity
• Conversion depends on wavelength-specific luminosity function
• Mathematically: Φ = K₃ ∫ P(λ)V(λ)dλ where:
  • P(λ) is spectral power distribution
  • V(λ) is luminosity function
  • K₃ is a constant

Definition of lumen:

• 1 lumen equals the luminous flux of a 1/683 watt monochromatic light source of wavelength 555 nm
• Alternatively: 1 watt of 555 nm light produces 683 lumens
• 555 nm corresponds to peak sensitivity of human vision under normal lighting conditions
• Basis for all photometric measurements that relate to human visual perception

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 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

A solid angle is a measure of the angular divergence of a cone in three-dimensional space:

• Mathematically defined as: ω = A/R²
• Where A is the area intercepted by the cone on a sphere of radius R centered at the apex of the cone
• Unit: steradian (sr)
• The total solid angle about a point is 4π steradians (complete sphere)
• Unlike plane angles measured in radians, solid angles measure how much of the "view" from a point is occupied by an object

Example: A cone that intercepts 1 m² on a sphere of radius 1 m subtends a solid angle of 1 steradian.

For a small area A on a surface tilted at angle θ from the normal:

• The solid angle is given by: ω = (A × cos θ)/R²
• Where:
  • A is the actual area of the surface
  • θ is the angle between the normal to the surface and the line to the observation point
  • R is the distance from the observation point to the surface
  • cos θ accounts for the apparent reduction in area due to the tilt (projected area)

This formula is essential in:

• Photometry when calculating illuminance from angled surfaces
• Radiative heat transfer calculations
• Astronomical observations of extended objects
• Computer graphics for rendering and ray tracing

Example: A 1 m² panel viewed from 10 m away at a 60° angle subtends a solid angle of (1 × cos 60°)/10² = 0.005 sr

The Bunsen photometer works on the principle of equal illuminance:

• When two screens are illuminated by separate light sources and appear equally bright, their illuminance is equal
• For two light sources with intensities I₁ and I₂ at distances d₁ and d₂ from the screens, the relationship is:
  • I₁/I₂ = (d₁/d₂)²

Operation:

1. Light sources S₁ and S₂ are placed at opposite ends of an optical bench
2. A screen with translucent center portion is positioned between them
3. The screen is moved until both sides appear equally bright
4. The distances are measured and the intensity ratio is calculated

Example: If a standard 16-candela lamp appears equally bright as an unknown source when the standard is 40cm from the screen and the unknown is 20cm from the screen, the unknown source has an intensity of 4 candela (16 × (20/40)²).

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).

Difference between perfectly diffused surface and point source:

Perfectly Diffused Surface:

• Follows Lambert's Cosine Law: I = I₀cosθ
• Intensity varies with angle from normal
• Maximum intensity occurs perpendicular to surface
• Radiates in hemisphere (solid angle of 2π steradians)
• Brightness appears the same from all viewing angles

Point Source:

• Emits radiation uniformly in all directions
• Intensity is constant regardless of angle: I = constant
• Radiates in full sphere (solid angle of 4π steradians)
• Follows the inverse square law for illuminance

Example: A flat ceiling LED panel (diffuse source) appears equally bright from different angles in a room, while a bare bulb (approximate point source) appears equally intense from all directions.