Problem 17.11
Light from a helium-neon laser (wavelenght = 633 nm) is used to illuminate two narrow slits. The interference pattern is observed on a screen 3.0 m behind the slits. Thirteen bright fringes are seen, spanning a distance of 52 mm .
Part A -
What is the spacing (in mm ) between the slits?
Express your answer using two significant figures.
Well, it seems like those slits are causing quite the interference, huh? Let's do some mathemagic to figure it out!
To find the spacing between the slits, we can use the formula:
spacing = (wavelength * distance) / number of fringes
Plugging in the values we have:
spacing = (633 nm * 3.0 m) / 13
Now, let's convert that wavelength from nanometers to millimeters:
633 nm = 0.633 mm
So we have:
spacing = (0.633 mm * 3.0 m) / 13
Simplifying further:
spacing = (1.899 mm * 3.0) / 13
spacing = 5.697 mm / 13
spacing ≈ 0.438 mm
So, the spacing between the slits is approximately 0.438 mm. Keep those fringes dancing!
To find the spacing between the slits, we can use the formula for the fringe spacing in Young's double-slit experiment:
d*sin(theta) = m*lambda
where:
- d is the spacing between the slits
- theta is the angle between the central maximum and the mth order fringe
- m is the order of the fringe (in this case, m = 1 since we are considering the first order fringe)
- lambda is the wavelength of the light
In this problem, we are given:
- lambda = 633 nm = 633 x 10^-9 m (converting from nm to m)
- the distance between the slits and the screen: L = 3.0 m
- the number of fringes observed: n = 13
- the span of the observed fringes on the screen: s = 52 mm = 52 x 10^-3 m
First, we need to find the angle between the central maximum and the first order fringe. We can use the small angle approximation and assume that sin(theta) is approximately equal to the tangent of theta:
sin(theta) ≈ tan(theta) = s / L
Let's calculate that:
sin(theta) ≈ tan(theta) = 52 x 10^-3 m / 3.0 m = 17.33 x 10^-3
Next, we can solve the equation for the fringe spacing to find d:
d = m * lambda / sin(theta)
d = 1 * 633 x 10^-9 m / (17.33 x 10^-3)
d ≈ 36.5 x 10^-6 m
Finally, we convert the fringe spacing from meters to millimeters:
d ≈ 36.5 x 10^-6 m * 1000 = 36.5 x 10^-3 mm
Therefore, the spacing between the slits is approximately 36.5 mm.
To find the spacing between the slits, we can use the equation for the fringe spacing in a double-slit interference pattern:
d*sin(theta) = m*λ
Where:
- d is the spacing between the slits
- theta is the angle of the fringe from the central maximum
- m is the order of the fringe (m = 0 for the central maximum)
- λ is the wavelength of light
In this case, we are given:
- lambda (wavelength) = 633 nm = 633 x 10^-9 m
- m = 13 (since there are thirteen bright fringes)
- The distance between the screen and the slits = 3.0 m
First, let's find the angle theta:
tan(theta) = (distance to fringe) / (distance to screen)
tan(theta) = (52 mm / 2) / (3.0 m)
tan(theta) = 0.0267
theta ≈ 1.53°
Now we can substitute these values into the equation to find the spacing d:
d*sin(theta) = m*λ
d*sin(1.53°) = 13 * 633 x 10^-9 m
d = (13 * 633 x 10^-9 m) / sin(1.53°)
Now let's calculate this value:
d = (13 * 633 x 10^-9 m) / sin(1.53°)
d ≈ 0.000001385 m
To convert this to millimeters, we multiply by 1000:
d ≈ 1.385 mm
Therefore, the spacing between the slits is approximately 1.385 mm.