Question

Light of wavelength \(653 \mathrm{nm}\) illuminates a slit. If the angle between the first dark fringes on either side of the central maximum is \(32.0^{\circ},\) What is the width of the slit?

Short answer

Question: Determine the width of the slit given a wavelength of light of 653 nm and an angle of 32.0 degrees between the first dark fringes on either side of the central maximum. Answer: The width of the slit is approximately \(2.46 \times 10^{-6}\) m or 2.46 \(\mu\)m.

Step-by-step solution

Determine the given values

We are given the following values: - Wavelength of light (\(\lambda\)): \(653 \ \text{nm}\) - Angle between the first dark fringes on either side of the central maximum: \(32.0^\circ\). We need to find the width of the slit (\(a\)).

Convert the wavelength of light to meters

We need to convert the given wavelength of light from nanometers to meters to work with SI units. We know that 1 nm = \(10^{-9}\) m: $$\lambda = 653 \ \text{nm} \cdot 10^{-9} \ \frac{\text{m}}{\text{nm}} = 6.53 \times 10^{-7} \ \text{m}$$

Calculate the angle of the first dark fringe from the central maximum

We are given the total angle between the two first dark fringes on either side of the central maximum (\(32.0^\circ\)). To find the angle of the first dark fringe from the central maximum (\(\theta\)), we can divide the total angle by 2: $$\theta = \frac{32.0^\circ}{2} = 16^\circ$$

Use the formula for the angular position of dark fringes to solve for the width of the slit

We can use the formula for the angular position of dark fringes in a single-slit diffraction pattern: $$\sin \theta = \frac{m \lambda}{a}$$ We know that the order of the first dark fringe (\(m\)) is 1, so we can plug in the known values and solve for the width of the slit (\(a\)): $$\sin 16^\circ = \frac{1 \cdot 6.53 \times 10^{-7} \ \text{m}}{a}$$ Now, we can solve for \(a\): $$a = \frac{1 \cdot 6.53 \times 10^{-7} \ \text{m}}{\sin 16^\circ} \approx 2.46 \times 10^{-6} \ \text{m}$$ Hence, the width of the slit is approximately \(2.46 \times 10^{-6}\) m or 2.46 \(\mu\)m.

Key concepts

Understanding Wavelength of Light

The wavelength of light is the distance between successive peaks (or troughs) of a wave. This is particularly crucial in the study of light behavior, as it determines many of the observable characteristics of light, including its color when it lies within the visible spectrum. When light interacts with objects close to its wavelength in size, this property becomes a key player in phenomena like diffraction.

For instance, the exercise provided deals with light of wavelength 653 nanometers (nm), which is a value found in the visible part of the spectrum, corresponding to a red hue. It's important to convert this wavelength into meters (\( 6.53 \times 10^{-7} \text{m} \)) when we want to incorporate it into mathematical equations used in physics, as it allows for consistency and accuracy in calculations. The wavelength is a fundamental component not just in predicting the color of light but also in understanding and calculating diffraction patterns.

Deciphering Diffraction Patterns

When we talk about diffraction patterns, we are generally referring to the series of light and dark bands produced when light waves bend, or 'diffract', around the edges of an object, like a slit. The patterns are tangible proof of the wave nature of light. In a single-slit diffraction setup, a series of dark and bright fringes are observed on a screen behind the slit.

These patterns are formed due to constructive and destructive interference of the light waves emanating from different parts of the slit. Constructive interference leads to bright fringes where the light waves amplify each other, while destructive interference gives rise to dark fringes where the light waves cancel each other out. The central maximum, the brightest of the fringes, occurs at the point where light does not have to bend. The exercise focuses on the position of the dark fringes, which is where the light waves destructively interfere and cancel out.

Locating the Angular Position of Dark Fringes

The angular position of dark fringes in single-slit diffraction is the angle at which these fringes appear relative to the central axis of the light pattern. This angle is vital in calculating the slit width because it directly relates to the spacing between the fringes and the wavelength of light used in the experiment. To find dark fringes, one can use the formula \(\sin(\theta) = \frac{m\lambda}{a}\), where \(\theta\) is the angle of the fringe, \(m\) is the order number of the fringe (1 for the first dark fringe), \(\lambda\) is the wavelength of light, and \(a\) is the width of the slit.

In our specific exercise, we have the angle between the first dark fringes on either side of the central maximum. To find the angle \(\theta\) for a single side, one must halve this total angle. Understanding the angular position and its application in the formula allows us to solve for unknowns, like the slit width in this case, providing a valuable insight into how light interacts with small openings and contributes to the broader field of wave optics.