Two slits are illuminated by a 536 nm light.The angle between the zeroth-order bright band at the center of the screen and the fourth-order bright band is 16.1 degrees. If the screen is 176 cm from the double-slit, how far apart is this bright band from the
central peak? Answer in units of cm.
What is the distance between the two slits? Answer in units of mm.
To calculate the distance between the bright band and the central peak, we need to use the equation for the position of the bright bands in a double-slit interference pattern:
y = (mλL) / d
where:
y is the distance between the central peak and the bright band,
m is the order of the bright band (4 in this case),
λ is the wavelength of light (536 nm = 536 × 10^(-9) m),
L is the distance between the screen and the double-slit (176 cm = 1.76 m),
and d is the distance between the slits.
Now, let's plug in the values and solve for y:
y = (4 × 536 × 10^(-9) × 1.76) / d
y = (2.26944 × 10^(-6)) / d
We also know that the angle between the central peak and the fourth-order bright band is 16.1 degrees. Using trigonometry, we can relate this angle to the distance y:
tan(angle) = y / L
tan(16.1) = y / 1.76
y = 1.76 × tan(16.1)
Now, we can set both expressions for y equal to each other and solve for d:
(2.26944 × 10^(-6)) / d = 1.76 × tan(16.1)
(2.26944 × 10^(-6)) = 1.76 × d × tan(16.1)
d = (2.26944 × 10^(-6)) / (1.76 × tan(16.1))
Now, we can calculate the value of d, which is the distance between the two slits:
d = (2.26944 × 10^(-6)) / (1.76 × tan(16.1))
To get the answer in units of cm, we multiply the calculated value of d by 10:
d = ((2.26944 × 10^(-6)) / (1.76 × tan(16.1))) × 10
Now, we have the value of d in mm. To convert it to cm, we divide it by 10:
d = ((2.26944 × 10^(-6)) / (1.76 × tan(16.1))) × 10 / 10
Now, we can calculate the value of d in mm and convert it to cm.