To determine the distance that separates the first-order maximum and the central maximum in a double-slit interference pattern, we'll use the following formula:
[tex]\[ y = \frac{m \lambda D}{d} \][/tex]
where:
- [tex]\( y \)[/tex] is the distance between the m-th order maximum and the central maximum,
- [tex]\( m \)[/tex] is the order of the maximum (for first-order, [tex]\( m = 1 \)[/tex]),
- [tex]\( \lambda \)[/tex] is the wavelength of the light,
- [tex]\( D \)[/tex] is the distance from the slits to the screen,
- [tex]\( d \)[/tex] is the distance between the two slits.
Given the values:
- [tex]\( \lambda = 5.8 \times 10^{-7} \)[/tex] meters (the wavelength of yellow light),
- [tex]\( d = 2.0 \times 10^{-4} \)[/tex] meters (the distance between the two slits),
- [tex]\( D = 2.0 \)[/tex] meters (the distance from the slits to the screen).
We need to find [tex]\( y \)[/tex] for the first-order maximum, so [tex]\( m = 1 \)[/tex]:
[tex]\[ y = \frac{1 \times 5.8 \times 10^{-7} \times 2.0}{2.0 \times 10^{-4}} \][/tex]
Let's simplify the expression:
First, multiply the numerator values:
[tex]\[ 1 \times 5.8 \times 10^{-7} \times 2.0 = 1.16 \times 10^{-6} \][/tex]
Next, divide by the denominator:
[tex]\[ y = \frac{1.16 \times 10^{-6}}{2.0 \times 10^{-4}} = \frac{1.16}{2.0} \times 10^{-2} \][/tex]
[tex]\[ y = 0.58 \times 10^{-2} \][/tex]
Converting [tex]\( 0.58 \times 10^{-2} \)[/tex] to scientific notation:
[tex]\[ y = 5.8 \times 10^{-3} \][/tex]
Thus, the distance separating the first-order maximum and the central maximum is:
[tex]\[ \boxed{5.8 \times 10^{-3} \, \text{meters}} \][/tex]
This matches option (3).
