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Chapter 25: Problem 75

Two slits separated by \(20.0 \mu \mathrm{m}\) are illuminated by light of wavelength \(0.50 \mu \mathrm{m} .\) If the screen is \(8.0 \mathrm{m}\) from the slits, what is the distance between the \(m=0\) and \(m=1\) bright fringes?

Short Answer

Expert verified
Answer: The distance between the central bright fringe (m=0) and the first bright fringe adjacent to the central fringe (m=1) is 0.20 m.

Step by step solution

01

Write down the double-slit interference formula

The interference formula for a double-slit experiment is given by: $$ \Delta y = \dfrac{m \lambda d}{a} $$ where \(\Delta y\) is the distance between the adjacent bright fringes, \(m\) is the fringe order (\(0,1,2,\dots\)), \(\lambda\) is the wavelength of the light, \(d\) is the distance between the screen and the slits, \(a\) is the distance between the slits.
02

Plug in the given values

We are given the distance between the slits (\(a = 20.0\,\mu m\)), the wavelength of the light (\(\lambda = 0.50\,\mu m\)), and the distance between the screen and the slits (\(d = 8.0\,m\)). We are looking for the distance between the \(m=0\) and \(m=1\) bright fringes.
03

Calculate the distance between the bright fringes

We need to find the difference in the fringe distances for \(m=0\) and \(m=1\). In other words, we want to calculate \(\Delta y(m=1) - \Delta y (m=0)\). Using the given values, we can plug them into the formula: $$ \begin{aligned} \Delta y (1) &= \dfrac{1 \times 0.50\,\mu m \times 8.0\,m}{20.0\,\mu m} \\ \Delta y (0) &= \dfrac{0 \times 0.50\,\mu m \times 8.0\,m}{20.0\,\mu m} \\ \end{aligned} $$
04

Solve for the distance between the bright fringes

Solving for the distances, we get: $$ \begin{aligned} \Delta y (1) &= \dfrac{4.0}{20.0\,\mu m} m = 0.20\, m \\ \Delta y (0) &= 0\, m \end{aligned} $$ Since the distance between the \(m=0\) and \(m=1\) bright fringes corresponds to \(\Delta y(1) - \Delta y(0)\), we find: $$ \Delta y = 0.20\, m - 0\, m = 0.20\, m $$ Hence, the distance between the \(m=0\) and \(m=1\) bright fringes is \(0.20\,m\).

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