Question

Suppose the distance between the slits in a double-slit experiment is \(2.00 \cdot 10^{-5} \mathrm{~m} .\) A beam of light with a wavelength of \(750 \mathrm{nm}\) is shone on the slits. What is the angular separation between the central maximum and the adjacent maximum? a) \(5.00 \cdot 10^{-2} \mathrm{rad}\) b) \(4.50 \cdot 10^{-2} \mathrm{rad}\) c) \(3.75 \cdot 10^{-2} \mathrm{rad}\) d) \(2.50 \cdot 10^{-2} \mathrm{rad}\)

Short answer

Answer: The angular separation between the central maximum and the adjacent maximum is approximately \(3.75 \cdot 10^{-2} \mathrm{rad}\).

Step-by-step solution

Convert the wavelength to meters

First, we need to convert the given wavelength of the light, 750 nm, to meters. 1 nm = \(10^{-9} m\), so 750 nm = \(750 \cdot 10^{-9} m\).

Substitute given values into the formula for the angular separation

Now substitute the given values for d and λ into the equation we derived for θ: $$ \theta = \sin^{-1} \left( \frac{1 \cdot (750 \cdot 10^{-9})}{2.00 \cdot 10^{-5}} \right) $$

Calculate the sine inverse

Evaluate the sine inverse to find the angular separation: $$ \theta = \sin^{-1} \left( \frac{750 \cdot 10^{-9}}{2.00 \cdot 10^{-5}} \right) \approx 0.0375 \, \text{rad} $$

Compare the result to the given options

Comparing our result to the given options, we can see that 0.0375 rad is closest to option c) \(3.75 \cdot 10^{-2} \mathrm{rad}\). The angular separation between the central maximum and the adjacent maximum is approximately \(3.75 \cdot 10^{-2} \mathrm{rad}\).

Key concepts

Wave-Particle Duality

One of the most fascinating and mind-bending concepts in physics is wave-particle duality, which forms the cornerstone of quantum mechanics. The double-slit experiment is often used to illustrate this phenomenon because it demonstrates how light and matter can exhibit both wave-like and particle-like properties.

When light, or electrons, are shone through two adjacent slits, they create a pattern on a screen that suggests interference—a distinctive characteristic of waves. Yet, when we try to measure the exact path of these particles through one slit, suddenly they behave like particles whose exact trajectories can be tracked. This dual behavior challenges our classical ideas about the nature of reality.

To make this concept relatable, imagine if you could be in two places at the same time when you walk through a doorway, only to become one person again when someone watches you. Wave-particle duality tells us that on a quantum level, the universe functions in this seemingly impossible way.

Interference Pattern

The interference pattern is what you get when two sets of waves, such as light waves in the double-slit experiment, overlap and combine. Imagine dropping two stones in a still pond at different points. The ripples from each stone will expand outwards and eventually intersect. Where the peaks of those ripples meet, they'll add together to make a higher peak; where a peak meets a trough, they'll cancel each other out to flatten the water. This is called 'constructive' and 'destructive' interference, respectively.

In the double-slit experiment, this type of interference creates a series of bright and dark bands on a screen. The pattern can tell scientists a lot about the light or particles used in the experiment, such as their wavelength. That's because the position and spacing of the bands are directly related to the wavelength, ensuring that the interference pattern acts as a wave 'signature' for the light or particles passing through the slits.

Angular Separation Physics

Angular separation in physics refers to the measure of the angle formed between two points as seen from an observer's point of view. Think of it as the measure of 'spread' between those two points in the sky, or in the case of the double-slit experiment, between two bright or dark fringes on the interference pattern.

The angular separation tells us how far apart objects appear to be. For example, in astronomy, it helps us determine the distance between two stars as seen from Earth. In the double-slit experiment's context, it helps us calculate the angle between the central bright fringe (known as the central maximum) and the first bright fringe adjacent to it (the first-order maximum). This is crucial for determining how the interference pattern will look at a particular distance from the slits and with a specific wavelength of light, as shown in the given exercise.

To bring it back to the exercise provided, the correct choice for the angular separation between the central and adjacent maxima, using the provided wavelength of light and distance between slits, would be option c), which aligns with the calculated angle of approximately 0.0375 radians.