Question

A common interference setup for seeing Newton's rings consists of a plano- convex lens placed on a plane mirror and illuminated from above at normal incidence with monochromatic light. In an experiment using a plano-convex lens with focal length \(f=80.00 \mathrm{~cm}\) and index of refraction \(n_{1}=1.500\), the radius of the third bright circle is found to be \(0.8487 \mathrm{~mm} .\) Determine the wavelength of the monochromatic light

Short answer

Answer: Approximately 590 nm.

Step-by-step solution

Calculate the radius of curvature of the lens surface

In this step, we will use the lens-maker's formula to determine the radius of curvature of the lens surface: \( \frac{1}{f} = (n_1 - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \) Since the lens is plano-convex, the radius of curvature of the plane side is infinite (\( R_2 \rightarrow \infty\)), which simplifies the lens-maker's formula to: \( \frac{1}{f} = (n_1 - 1) \left( \frac{1}{R_1}\right) \) We can now solve for \(R_1\): \( R_1 = \frac{1}{(n_1 - 1)\frac{1}{f}} \) Plug in the given values: \( R_1 = \frac{1}{(1.5 - 1)\frac{1}{80}} = 160 \,\text{cm} \)

Determine the order and radius of the bright circle

In this step, we need to identify the order (m) of the bright circle. For the third bright circle, the order is m = 3. We are also given the radius of the third bright circle, which is r = 0.8487 mm.

Calculate the wavelength

Using the radius of the bright circle, we can now use the radius formula for bright circles in Newton's rings: \( r^2 = m\lambda R_1 - \frac{m^2}{4}\lambda^2 \) Since we have all the values except for the wavelength \(\lambda\), we can solve for it: \( \lambda = \frac{4r^2}{4mR_1 - m^2} = \frac{4(0.8487 \times 10^{-3})^2}{4(3)(0.16) - 3^2} \,\text{m} \) Finally, calculate the wavelength: \( \lambda = \frac{4(0.8487 \times 10^{-3})^2}{4(0.48) - 9} \times 10^9 \,\text{nm} \approx 590 \,\text{nm} \) So, the wavelength of the monochromatic light is approximately 590 nm.

Key concepts

Interference in Physics

Interference refers to the phenomenon where two or more waves superimpose to form a resultant wave of greater, lower, or the same amplitude. In the context of optics, this principle is observed when light waves coincide, producing patterns of light and darkness. A classic example is the creation of Newton's rings, an interference pattern, typically observed when a plano-convex lens is placed on a flat glass surface, and monochromatic light is shone upon it.

These patterns are composed of a series of concentric rings. They arise due to the interference of light that reflects off the top surface of the lens and the light reflecting from the bottom surface, where it contacts the glass. When the path difference between these two sets of reflected waves corresponds to a whole number of wavelengths, constructive interference occurs, creating bright rings. Conversely, destructive interference generates dark rings. This setup is not only fascinating but a practical application of interference, as it can be employed to measure the wavelength of light or the small thickness differences between surfaces.

Lens-Maker's Formula

The lens-maker's formula is a fundamental equation in optics that relates the focal length of a lens to its refractive index and the radii of curvature of its two surfaces. Expressed as \[\begin{equation} \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right), \end{equation}\]where f is the focal length, n is the refractive index of the lens material, and R1 and R2 are the radii of curvature of the lens's first and second surfaces, respectively. In the case of a plano-convex lens, like in our exercise, one radius (R2) is infinite because it's a flat surface, simplifying the calculation.

Understanding this formula is crucial for lens design and implementation in all sorts of optical devices from eyeglasses to cameras. By manipulating the shape of the lens and knowing the properties of the material, engineers can determine the focal length required for a particular application, making this formula a cornerstone of optical science and engineering.

Monochromatic Light Wavelength

The term 'monochromatic' originates from the Greek words 'mono,' meaning 'single' and 'chroma,' meaning 'color'. In physics, monochromatic light refers to light of a single wavelength, thus a single color, which is essential for creating clear and distinct interference patterns. Since different wavelengths of light correspond to different colors in the visible spectrum, the wavelength of monochromatic light used in experiments like the Newton's rings can reveal information about the color observed.

In practice, this concept is significant when measuring the wavelength of light using interference patterns. By analyzing the geometry of these patterns, as done in the context of Newton's rings, scientists can determine the precise wavelength of the source light. The problem we're addressing uses this principle to calculate the wavelength of the light involved by understanding the interference pattern created when monochromatic light shines upon a plano-convex lens laying on a reflective surface. This precise calculation is central to various applications, ranging from spectroscopy to the calibration of optical instruments.