Coherent light of frequency 6.32 × 10¹⁴ Hz passes through two thin slits and falls on a screen 85.0 cm away. You observe that the third bright fringe occurs at ±3.11 cm on either side of the central bright fringe. (a) How far apart are the two slits? (b) At what distance from the central bright fringe will the third dark fringe occur?

Verified step by step guidance

1. Identify the given quantities: frequency of light \(f = 6.32 \times 10^{14}\) Hz, distance to screen \(L = 85.0\) cm, and position of the third bright fringe \(y_3 = \pm 3.11\) cm. Convert all distances to meters for consistency in SI units.
2. Calculate the wavelength \(\lambda\) of the light using the relation between frequency and wavelength: \(\lambda = \frac{c}{f}\), where \(c = 3.00 \times 10^8\) m/s is the speed of light in vacuum.
3. Use the formula for the position of bright fringes in a double-slit interference pattern: \(y_m = \frac{m \lambda L}{d}\), where \(m\) is the fringe order (for the third bright fringe, \(m=3\)), \(L\) is the distance to the screen, and \(d\) is the slit separation. Rearrange this formula to solve for \(d\): \(d = \frac{m \lambda L}{y_m}\).
4. For part (b), recall that dark fringes occur at positions given by \(y_{dark} = \frac{(m + \frac{1}{2}) \lambda L}{d}\), where \(m\) is the dark fringe order starting from zero. Since we want the third dark fringe, set \(m=2\) (because the first dark fringe corresponds to \(m=0\)), and calculate \(y_{dark}\) using the previously found \(d\) and \(\lambda\).
5. Interpret the results: the slit separation \(d\) found in part (a) tells how far apart the two slits are, and the position \(y_{dark}\) from part (b) gives the distance from the central bright fringe to the third dark fringe on the screen.

Key Concepts

Double-Slit Interference

Double-slit interference occurs when coherent light passes through two closely spaced slits, producing a pattern of bright and dark fringes on a screen due to constructive and destructive interference of light waves. The position of bright fringes depends on the slit separation, wavelength, and distance to the screen.

Relationship Between Fringe Position and Wavelength

The position of bright fringes (maxima) in a double-slit experiment is given by y = (mλL)/d, where m is the fringe order, λ is the wavelength, L is the distance to the screen, and d is the slit separation. This formula links the physical setup to the observed interference pattern.

Dark Fringe Position in Double-Slit Interference

Dark fringes (minima) occur where destructive interference happens, located at positions y = ((m + 0.5)λL)/d for integer m. These positions lie between the bright fringes and are essential for determining the full interference pattern on the screen.