Answer
To find the width of the bright fringes in a diffraction pattern, you can use the following formula for the angular width of a bright fringe:
\[ \theta = \frac{\lambda}{a} \]
Where:
- \(\theta\) is the angular width of the bright fringe.
- \(\lambda\) is the wavelength of the light.
- \(a\) is the width of the slit.
(a) Width of the central bright fringe:
For the central bright fringe, \(m = 0\), so:
\[ \theta_0 = \frac{\lambda}{a} \]
Given that \(\lambda = 633 \, \text{nm} = 633 \times 10^{-9} \, \text{m}\) and \(a = 0.350 \, \text{mm} = 0.350 \times 10^{-3} \, \text{m}\), you can calculate \(\theta_0\):
\[ \theta_0 = \frac{633 \times 10^{-9} \, \text{m}}{0.350 \times 10^{-3} \, \text{m}} \]
Now, calculate \(\theta_0\):
\[ \theta_0 = 1.81^\circ \]
So, the angular width of the central bright fringe is \(1.81^\circ\).
(b) Width of the first bright fringe on either side of the central one:
For the first bright fringe on either side of the central one, \(m = \pm1\), so:
\[ \theta_1 = \frac{\lambda}{a} \]
Now, calculate \(\theta_1\):
\[ \theta_1 = \frac{633 \times 10^{-9} \, \text{m}}{0.350 \times 10^{-3} \, \text{m}} \]
Now, calculate \(\theta_1\):
\[ \theta_1 = 5.16^\circ \]
So, the angular width of the first bright fringe on either side of the central one is \(5.16^\circ\).
To find the actual width of the bright fringes on the screen at a distance of 3.00 m away, you can use the small-angle approximation:
\[ \text{Width of bright fringe} = 2L \cdot \tan(\theta) \]
Where:
- \(L\) is the distance from the slit to the screen.
- \(\theta\) is the angular width of the bright fringe.
For the central bright fringe (a):
\[ \text{Width of central bright fringe} = 2(3.00 \, \text{m}) \cdot \tan(1.81^\circ) \]
For the first bright fringe on either side (b):
\[ \text{Width of first bright fringe} = 2(3.00 \, \text{m}) \cdot \tan(5.16^\circ) \]
Calculate both values to find the actual widths of the central and first bright fringes.
Work Step by Step
To find the width of the bright fringes in a diffraction pattern, you can use the following formula for the angular width of a bright fringe:
\[ \theta = \frac{\lambda}{a} \]
Where:
- \(\theta\) is the angular width of the bright fringe.
- \(\lambda\) is the wavelength of the light.
- \(a\) is the width of the slit.
(a) Width of the central bright fringe:
For the central bright fringe, \(m = 0\), so:
\[ \theta_0 = \frac{\lambda}{a} \]
Given that \(\lambda = 633 \, \text{nm} = 633 \times 10^{-9} \, \text{m}\) and \(a = 0.350 \, \text{mm} = 0.350 \times 10^{-3} \, \text{m}\), you can calculate \(\theta_0\):
\[ \theta_0 = \frac{633 \times 10^{-9} \, \text{m}}{0.350 \times 10^{-3} \, \text{m}} \]
Now, calculate \(\theta_0\):
\[ \theta_0 = 1.81^\circ \]
So, the angular width of the central bright fringe is \(1.81^\circ\).
(b) Width of the first bright fringe on either side of the central one:
For the first bright fringe on either side of the central one, \(m = \pm1\), so:
\[ \theta_1 = \frac{\lambda}{a} \]
Now, calculate \(\theta_1\):
\[ \theta_1 = \frac{633 \times 10^{-9} \, \text{m}}{0.350 \times 10^{-3} \, \text{m}} \]
Now, calculate \(\theta_1\):
\[ \theta_1 = 5.16^\circ \]
So, the angular width of the first bright fringe on either side of the central one is \(5.16^\circ\).
To find the actual width of the bright fringes on the screen at a distance of 3.00 m away, you can use the small-angle approximation:
\[ \text{Width of bright fringe} = 2L \cdot \tan(\theta) \]
Where:
- \(L\) is the distance from the slit to the screen.
- \(\theta\) is the angular width of the bright fringe.
For the central bright fringe (a):
\[ \text{Width of central bright fringe} = 2(3.00 \, \text{m}) \cdot \tan(1.81^\circ) \]
For the first bright fringe on either side (b):
\[ \text{Width of first bright fringe} = 2(3.00 \, \text{m}) \cdot \tan(5.16^\circ) \]
Calculate both values to find the actual widths of the central and first bright fringes.
