Speed of Light

For no net angular dispersion, the ratio of refracting angles (A'/A) must equal:

$\frac{A'}{A} = \frac{2\mu_v - \mu_r}{\mu'_v - \mu'_r}$

Where:

• μᵥ and μᵣ are refractive indices for violet and red light in crown glass
• μ'ᵥ and μ'ᵣ are refractive indices for violet and red light in flint glass

This ensures that the dispersive effects of both prisms exactly cancel each other out while still allowing for net deviation.

In Foucault's method:

1. Light reflects from a rotating mirror to a fixed concave mirror and back
2. During travel time (t = 2R/c), the mirror rotates by angle θ = ωt
3. This causes a measurable displacement (s) in the returning beam

The speed of light equation is derived by:

• The angle of rotation: θ = ωt = 2Rω/c
• The measured displacement: s = 2Rθ·a/(R+b)
• Substituting and solving for c:

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

Where R is mirror radius, ω is angular speed, a is the distance from lens to source, b is distance from mirror to lens, and s is the measured displacement.

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.

For no net deviation in the yellow ray (while still allowing dispersion):

$\frac{A'}{A} = \frac{2\mu_y - 1}{\mu'_y - 1}$

Where:

• A' is the refracting angle of the second prism
• A is the refracting angle of the first prism
• μy is the refractive index of the first prism for yellow light
• μ'y is the refractive index of the second prism for yellow light

This creates a system where:

• The yellow ray passes through undeviated
• Other wavelengths still experience dispersion
• The system acts as a direct-vision spectroscope where colors separate without overall bending of light

This principle is used in direct-vision spectroscopes to produce a spectrum along the original light direction.

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.

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.