a) Describe Newton's ring experiment and derive expression for wavelength of light. b) Describe Foucault's method of determining the speed of light.

a) Newton’s Ring Experiment and Derivation of Wavelength Expression

Description: Newton’s rings are concentric bright and dark interference fringes formed when a plano-convex lens is placed on a flat glass plate. A monochromatic light source (e.g., sodium lamp) illuminates the setup from above, and the interference pattern is observed through a microscope or directly on a screen.

Setup:

• A plano-convex lens (radius of curvature $R$) is placed on a flat glass plate.
• A thin air film is formed between the lens and the plate.
• Light reflects from the top surface of the air film and the bottom surface (glass plate), producing interference.

Interference Condition:

• For dark rings (destructive interference), the path difference between the two reflected rays is an odd multiple of half the wavelength: $$2t=(m+\frac{1}{2})\lambda$$ where:

  • $t$ = thickness of the air film at a point,
  • $m$ = order of the fringe (0, 1, 2, ...),
  • $\lambda$ = wavelength of light.

• The thickness $t$ of the air film at a distance $r$ from the center is given by: $$t=\frac{r^2}{2R}$$ (derived from geometry: $R^2=(R-t{)}^2+r^2$, assuming $t\ll R$).

Derivation of Wavelength: Substitute $t$ into the interference condition: $$2\left(\frac{r^2}{2R}\right)=\left(m+\frac{1}{2}\right)\lambda$$ Simplify: $$\frac{r^2}{R}=\left(m+\frac{1}{2}\right)\lambda$$ For the $m$-th dark ring, the radius $r_m$ is measured. Rearranging for $\lambda$: $$\lambda =\frac{r_m^2}{R\left(m+\frac{1}{2}\right)}$$ If the radii of the $m$-th and $(m+k)$-th dark rings are $r_m$ and $r_{m+k}$, respectively, then: $$\lambda =\frac{r_{m+k}^2-r_m^2}{kR\left(m+\frac{1}{2}+\frac{k}{2}\right)}$$ This allows experimental determination of $\lambda$ by measuring $r$ and knowing $R$.

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b) Foucault’s Method for Determining the Speed of Light

Principle: Foucault used a rotating mirror to measure the time taken by light to travel a known distance, thereby calculating its speed. The key idea is that light takes a finite time to traverse a path and return, causing a measurable angular shift in the reflected beam.

Setup:

1. A light source (S) emits a beam toward a partially silvered mirror (M₁), which splits the beam.
2. The reflected beam travels to a rotating mirror (M₂) placed at a distance $D$.
3. After reflection from $M_2$, the beam travels to a fixed mirror (M₃) at a distance $L$ from $M_2$.
4. The beam returns to $M_2$, reflects again, and travels back to $M_1$.
5. The final reflected beam is observed in a telescope or on a scale.

Working:

• When $M_2$ is stationary, the returning beam retraces its path and is reflected back to the source.
• If $M_2$ rotates at angular speed $\omega$ (rad/s), the beam is deflected by an angle $\theta =\omega t$, where $t$ is the total time taken for the round trip.
• The total distance traveled by light is $2D+2L$, so: $$t=\frac{2(D+L)}{c}$$ where $c$ is the speed of light.
• The angular deflection $\theta$ is measured as the displacement $d$ on a scale at distance $D^{\prime }$: $$\theta =\frac{d}{D^{\prime }}$$
• Equating the two expressions for $\theta$: $$\omega t=\frac{d}{D^{\prime }}⟹\omega \left(\frac{2(D+L)}{c}\right)=\frac{d}{D^{\prime }}$$
• Solving for $c$: $$c=\frac{2\omega (D+L)D^{\prime }}{d}$$
• Foucault measured $\omega$, $D$, $L$, $D^{\prime }$, and $d$ to compute $c\approx 2.98\times 10^8\text{m/s}$, close to the modern value.

Advantages:

• More accurate than earlier methods (e.g., Römer’s astronomical method).
• Allowed precise laboratory measurement of $c$.