Question

Determine the minimum thickness of a soap film \((n=1.32)\) that would produce constructive interference when illuminated by light of wavelength of \(550 . \mathrm{nm} .\)

Short answer

Answer: The minimum thickness of the soap film is approximately 208.3 nm.

Step-by-step solution

Write down the formula for constructive interference in thin films

The formula for constructive interference in thin films is given by: $$2\cdot n \cdot t = m\cdot \lambda$$ where: - \(n\) is the refractive index of the thin film, - \(t\) is the thickness of the thin film, - \(m\) is the order of interference (an integer), and - \(\lambda\) is the wavelength of light. We need to find the minimum thickness, which corresponds to the first order of interference \((m = 1)\). Therefore, we can rewrite our formula as: $$2\cdot n \cdot t = \lambda$$

Plug in given values

Now plug in the given values for the refractive index \((n = 1.32)\) and the wavelength of light \((\lambda = 550 \,\text{nm})\) into the formula: $$2\cdot 1.32 \cdot t = 550\, \text{nm}$$

Solve for the thickness of the soap film

Now, we need to isolate the variable \(t\) (the thickness) by dividing both sides of the equation by \(2\cdot 1.32\): $$ t = \frac{550\, \text{nm}}{2\cdot 1.32}$$ Calculate the value of \(t\): $$ t \approx 208.3 \,\text{nm}$$

State the result

The minimum thickness of the soap film that would produce constructive interference when illuminated by light of wavelength of \(550 \,\text{nm}\) is approximately \(208.3\, \text{nm}\).

Key concepts

Thin Film Interference

Thin film interference is a phenomenon where light waves reflected off the two sides of a thin film, such as soap bubbles or oil on water, interfere with each other. This type of interference can produce a spectrum of colors when light is shone on the film.

When light encounters a thin film, some of it is reflected off the top surface while some enters the film, gets reflected off the bottom surface, and exits out again. The light wave that enters the film experiences a change in speed due to the different optical density of the film material, known as its refractive index.

As the waves emerge from the film, they either constructively or destructively interfere with each other, depending on the thickness of the film, the wavelength of light, and the refractive index. Constructive interference, which is the focus of our problem, occurs when the path difference between two waves is an integer multiple of the wavelength, leading to a bright or colored appearance.

Refractive Index

The refractive index of a material, often denoted by the symbol 'n', is a measure of how much it reduces the speed of light. Specifically, it's the ratio of the speed of light in a vacuum to the speed of light in the material. Each material has a unique refractive index.

For example, in our soap film exercise, the refractive index is given as 1.32. This means that light travels 1.32 times slower in the soap film than it does in a vacuum. The refractive index is crucial in determining how much the light wave bends when entering a new medium and plays a vital role in how the waves interfere with each other inside the thin film.

Wavelength of Light

The wavelength of light, represented by the Greek letter lambda \(\lambda\), is the distance between successive crests (or troughs) of a light wave. Wavelength is typically measured in units of meters, with subunits such as nanometers (nm) used for very small distances.

In our exercise, the wavelength of light is given as 550 nm. Understanding the wavelength is essential because it determines the interference pattern within the thin film. Different wavelengths of light interfere differently due to their unique path differences within the film, which is why thin films often display a variety of colors.

Interference in Physics

Interference in physics refers to the phenomenon that occurs when two or more waves overlap and combine to form a new wave pattern. This can lead to areas of increased amplitude (constructive interference) and areas of reduced or zero amplitude (destructive interference).

The principle of superposition governs this pattern formation; the resultant wave at any point is the sum of the individual waves' amplitudes at that point. In the case of our problem involving a soap film, constructive interference is sought, which means finding the condition for the crests of the waves to coincide, reinforcing each other, and producing a maximum in the interference pattern.