Speed of Light

In Foucault's method, the key optical relationship is:

$\frac{s}{2R\Delta\theta} = \frac{a}{R+b}$

This applies the lens magnification principle where:

• The displacement s (image shift) relates to the angular shift 2R·Δθ (object shift)
• The ratio equals the magnification produced by the lens
• The magnification equals ratio of image distance (a) to object distance (R+b)

When the mirror rotates by angle Δθ during time Δt = 2R/c: $\Delta\theta = \omega\Delta t = \frac{2R\omega}{c}$

Substituting and solving yields the speed of light equation: $c = \frac{4R^2\omega a}{s(R+b)}$

Fundamental principle: The speed of light in vacuum is invariant - it has the same value in all inertial reference frames regardless of the observer's motion.

This led to Einstein's special theory of relativity and a complete revision of our concepts of space and time.

Modern impact on length definition:

• Since 1983, length is defined in terms of light speed
• 1 meter = distance light travels in 1/299,792,458 seconds
• The speed of light is now exactly 299,792,458 m/s by definition
• When measuring time for light to travel between two points, we are effectively measuring distance, not speed
• Prior to 1983, length was defined independently, allowing speed of light to be measured experimentally

This invariance of light speed represents one of the most profound shifts in our understanding of physics.

For thin prisms with similar orientation:

1. Angular dispersion for a single prism: ω = A(μᵥ - μᵣ) Where (μᵥ - μᵣ) is the difference in refractive indices for violet and red light

2. For two prisms with similar orientation, dispersions add: ω = A₁Δμ₁ + A₂Δμ₂ Where Δμ = (μᵥ - μᵣ) for each material

3. Example calculation (from problem 11b): For prisms with A₁ = 5.3°, Δμ₁ = 0.014, A₂ = 3.7°, Δμ₂ = 0.024: ω = 5.3° × 0.014 + 3.7° × 0.024 ω = 0.0742 + 0.0888 ω = 0.163°

This is much larger than when the prisms are oppositely directed (0.0146°), demonstrating how similar orientation increases dispersion.

Galileo's method failed because:

• It relied on human reaction time to cover/uncover lanterns when seeing light signals
• The speed of light is too large for human reflexes to measure accurately
• For a 15 km separation, light travels the round trip in only 0.0001 seconds
• Human reaction times are orders of magnitude slower (typically 0.1-0.3 seconds)

First successful measurements:

1. Astronomical methods (1676): Olaf Roemer measured irregularities in the eclipse times of Jupiter's moons, yielding approximately 2.1 × 10⁸ m/s
2. First terrestrial measurement (1849): Fizeau's toothed wheel experiment
3. Improved terrestrial methods: Foucault's rotating mirror (1862) and Michelson's refinements (1879)

Roemer's astronomical approach demonstrated that light had a finite speed, contradicting the prevailing belief that light propagation was instantaneous.

Current impact of meter definition on light speed measurements:

• Since 1983, the speed of light is defined as exactly 299,792,458 m/s
• This definition fixes the meter as the distance light travels in 1/299,792,458 seconds
• Modern "measurements" of light speed are actually calibrations of distance
• When timing light travel, we're effectively measuring distance, not speed
• Light speed is now a defined constant, not an experimentally determined value

Key implications:

1. No experiment can "measure" light speed differently than the defined value
2. Improved precision comes in time measurements, not speed determination
3. Distance measurements are now tied to time standards
4. This creates a unified framework where c is a fundamental constant
5. Relativity principles are built into our measurement system

This represents a fundamental shift from pre-1983 when length was defined independently (prototype meter bar) and light speed was measured experimentally.

For Fizeau's method, the speed of light is calculated as:

$c = 4Dnν$

Where:

• c = speed of light (m/s)
• D = distance from the wheel to the distant mirror (m)
• n = number of teeth in the toothed wheel
• ν = frequency of rotation (revolutions per second)

This equation derives from:

• The angle between teeth: θ = 2π/2n
• Time for wheel to rotate by this angle: t = θ/ω = π/(nω)
• During this time, light travels distance 2D
• Angular velocity ω = 2πν

The formula works because when the image disappears, the light's round-trip time precisely matches the time needed for the wheel to rotate from a gap to the adjacent tooth.

Foucault's method works by:

• Using a rotating mirror (M₁) to reflect light to a fixed concave mirror (M₂) and back
• During the light's travel time, the rotating mirror turns slightly, causing returning light to deflect
• This deflection creates a measurable shift in the returning beam's position

The fundamental principle: The shift in beam position (s) is proportional to light's travel time, allowing calculation of light speed using the formula:

$c = \frac{4R^2\omega a}{s(R+b)}$

Where:

• R = radius of the concave mirror
• ω = angular speed of the rotating mirror
• a = distance from lens to source
• b = distance from rotating mirror to lens
• s = shift in beam position

This method improved upon earlier techniques by allowing laboratory measurements in confined spaces.

Derivation of Foucault's formula for speed of light:

1. Define key variables:

   • θ = angle rotated by mirror during light travel
   • ω = angular speed of mirror
   • t = time for light to travel from M₁ to M₂ and back
   • R = radius of concave mirror
   • s = shift in position of returned beam
   • a = distance from lens to source
   • b = distance from rotating mirror to lens

2. Core relationships:

   • Time for light travel: t = 2R/c
   • Mirror rotation during this time: θ = ωt = ω(2R/c)
   • Optical displacement: OO' = R·(2θ) = 2Rθ

3. Apply magnification principle:

   • The lens forms an image with magnification = a/(R+b)
   • Therefore: s/(2Rθ) = a/(R+b)

4. Substitute θ = ω(2R/c):

   • s/[2R·ω(2R/c)] = a/(R+b)
   • s/[4R²ω/c] = a/(R+b)

5. Solve for c: $c = \frac{4R^2\omega a}{s(R+b)}$

This formula allows calculation of light speed by measuring all quantities on the right side.

Experimental Setup & Mechanism

• Light passes through a gap in a rapidly rotating toothed wheel
• Travels distance D to a mirror and returns (total path 2D)
• Image disappears at specific rotation speeds when:
  • During the light's travel time (t = 2D/c), the wheel rotates precisely enough (θ = 2π/2n) for a tooth to replace the gap
  • Returning light is blocked by the tooth that has rotated into position

Mathematical Calculation

The speed of light can be determined using: $c = \frac{2D \cdot n\omega}{\pi} = 4Dn\nu$ Where:

• D = distance to mirror
• n = number of teeth on wheel
• ω = angular velocity (radians/time)
• ν = rotation frequency (revolutions/time)

Scientific Significance

1. Provided first terrestrial (non-astronomical) measurement of light's speed
2. Demonstrated that light travels at a finite speed (not instantaneously)
3. Supported wave theory of light over corpuscular theory
4. Yielded results close to modern value (299,792,458 m/s)
5. Laid groundwork for understanding relativistic effects related to light propagation

Key Insight

The experiment's elegance lies in converting the time domain (light's travel time) into the spatial domain (angular rotation of a physical object), making an extremely fast phenomenon measurable with 19th century technology.

Key Innovation

• Rotating mirror system: Converted time measurement into spatial measurement (beam shift)
• This fundamental change allowed for precise measurements using 19th-century technology

Advantages over Fizeau's Method

1. Laboratory Implementation:

   • Required much less space (could be performed inside a laboratory)
   • Eliminated need for kilometers of open space
   • Enabled controlled experimental conditions for higher precision

2. Measurement Capabilities:

   • Allowed measurement of light speed in different transparent media by placing materials between mirrors
   • Produced better intensity of final image (less dimming)
   • Achieved higher precision with smaller equipment

Historical Significance

• Disproved Newton's Corpuscular Theory by directly demonstrating that light travels slower in dense media than in vacuum
• Provided crucial empirical evidence in the wave-particle debate about light's nature
• Represented a major advancement in experimental physics methodology

This method transformed our understanding of light's properties and established measurement techniques that influenced future optical experiments.