Question

Coherent monochromatic light passes through parallel slits and then onto a screen that is at a distance \(L=2.40 \mathrm{~m}\) from the slits. The narrow slits are a distance \(d=2.00 \cdot 10^{-5} \mathrm{~m}\) apart. If the minimum spacing between bright spots is \(y=6.00 \mathrm{~cm},\) find the wavelength of the light.

Short answer

The wavelength of the coherent monochromatic light is 500 nm.

Step-by-step solution

Identify the given values

We are given the following values: - Distance between the slits and the screen, \(L = 2.40 \mathrm{~m}\) - Distance between the slits, \(d = 2.00 \cdot 10^{-5} \mathrm{~m}\) - Minimum spacing between bright spots, \(y = 6.00 \mathrm{~cm} = 0.06 \mathrm{~m}\) (converted to meters) Since we're given the minimum spacing between bright spots, this implies that we are working with the first-order bright spots, which means \(m=1\).

Rearrange the formula for constructive interference

Our goal is to find the wavelength of the light, \(\lambda\). We can rearrange the formula for constructive interference in a double-slit interference setup to make \(\lambda\) the subject: $$\lambda = \frac{y_md}{mL}$$

Plug in the given values and calculate the wavelength

Now we can substitute the given values into the rearranged formula to find the wavelength of the light: $$\lambda = \frac{(0.06\mathrm{~m})(2.00 \cdot 10^{-5}\mathrm{~m})}{(1)(2.40\mathrm{~m})}$$

Solve for the wavelength

Carry out the calculation: $$\lambda = \frac{(0.06)(2.00 \cdot 10^{-5})}{2.40} = 5.0 \cdot 10^{-7} \mathrm{~m}$$ The wavelength of the light is \(5.0 \cdot 10^{-7}\mathrm{~m}\), or \(500 \mathrm{~nm}\).

Key concepts

Constructive Interference

When dealing with the wonders of light and its interactions, constructive interference is a fundamental principle that allows us to understand patterns like the dazzling array of colors in a soap bubble or the intricate fringes in a double-slit experiment. Constructive interference occurs when two waves meet in such a way that their crests (the highest points of the waves) align with each other, resulting in a wave of increased amplitude. Imagine clapping your hands as someone else claps at the exact same moment, creating a louder sound—that's a simple way to visualize how constructive interference amplifies waves.

Specifically, in the realm of light passing through slits, when coherent waves pass through multiple slits and emerge on the other side, their waves can combine. If they're in phase, meaning their wave peaks match up, they constructively interfere and reinforce each other, leading to bright spots, or maxima, on a screen. The location of these bright spots is determined by the wavelength of light and the geometry of the slits, playing into a precise calculation that gives us a symmetrical and predictable interference pattern.

Monochromatic Light

What do scientists mean when they talk about monochromatic light? Essentially, it's all about color and purity. Monochromatic light consists of one color or, more technically, one single wavelength of light. This purity is key for conducting precise experiments, such as the double-slit interference. If we were to use ordinary white light, which is a mix of all the colors (wavelengths), the result would be a confusing and overlapping pattern. But with monochromatic light, we get clean and distinguishable bright and dark regions due to interference.

To produce monochromatic light, lasers are often used because they emit light that is not only monochromatic but also coherent, meaning the light waves are in phase and have a fixed relationship to one another. This coherence is crucial in creating well-defined interference patterns, as it ensures that the light waves can interfere constructively or destructively in a controlled and repeatable manner.

Wavelength Calculation

The wavelength calculation in a double-slit interference scenario is an elegant choreography of mathematics and physics that brings these abstract concepts into sharp relief. The wavelength of light determines its color in the spectrum and plays a pivotal role in interference patterns. To calculate it, we use a formula derived from the double-slit setup parameters, which looks at the distance between slits (\(d\)), the order of the bright spot (\(m\break \

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                  To excerpt the magic numeral—the wavelength—we substitute our known values into the formula and unravel the math. This not only gives us the wavelength but also reinforces our grasp of how light waves work and interact at a fundamental level. Through calculations like these, students can connect abstract equations with tangible realities, deepening their comprehension of the subject at hand.