A student holds a laser that emits light of wavelength 641.5 nm. The laser beam passes though a pair of slits separated by 0.400 mm, in a glass plate attached to the front of the laser. The beam then falls perpendicularly on a screen, creating an interference pattern on it. The student begins to walk directly toward the screen at 3.00 m/s. The central maximum on the screen is stationary. Find the speed of the 50th-order maxima on the screen.
A particle at the Large Hadron collider in CERN is moving at 0.99 c as it emerges from a collision event.
A) If these particles live 2.0 microseconds on average when they are at rest, how long would you expect this particle to last in the reference frame of the physicist measuring the particle?
To find the speed of the 50th-order maxima on the screen, we need to calculate the change in position of the maxima as the student walks towards the screen. We can use the concept of the Doppler effect to find this change in position.
The Doppler effect describes the change in frequency or wavelength of a wave observed by an observer who is moving relative to the source of the wave. In this case, the observer is the student, and the source of the wave is the laser emitting light of wavelength 641.5 nm.
The formula for the Doppler effect with relative motion between the source of the wave and the observer is given by:
Δλ/λ₀ = v/c
where Δλ is the change in wavelength observed, λ₀ is the original wavelength, v is the velocity of the observer, and c is the speed of light.
In this scenario, the change in wavelength observed by the student is related to the change in position of the maxima on the screen. The change in position can be calculated using the formula:
Δx = d * (Δλ/λ₀)
where Δx is the change in position, d is the separation between the slits (0.400 mm), Δλ is the change in wavelength observed, and λ₀ is the original wavelength (641.5 nm).
Now, substitute the known values into the equation:
Δx = (0.400 mm) * (Δλ/λ₀) ...(1)
We need to find the speed of the 50th-order maxima, so let's assume that at the beginning, the 50th-order maxima is at position x. As the student walks towards the screen, the position of the 50th-order maxima changes to a new position x + Δx.
To find the speed of the 50th-order maxima, we need to find the time it takes for the student to walk from the original position to the new position:
Δt = Δx / v
where Δt is the change in time, Δx is the change in position, and v is the velocity of the student (3.00 m/s).
Substituting the value of Δx from equation (1) into the equation for Δt, we get:
Δt = [(0.400 mm) * (Δλ/λ₀)] / v
Now, we have the change in time Δt. The speed of the 50th-order maxima can be calculated by dividing the change in position Δx by the change in time Δt:
Speed of the 50th-order maxima = Δx / Δt
Substituting the values, we get:
Speed of the 50th-order maxima = [(0.400 mm) * (Δλ/λ₀)] / [(0.400 mm) * (Δλ/λ₀) / v]
Simplifying this expression, we get:
Speed of the 50th-order maxima = v
Therefore, the speed of the 50th-order maxima on the screen is equal to the velocity of the student, which is 3.00 m/s.