Diffraction grating - have answer and working, but don't understand it

[karnten07]

Homework Statement

A sodium lamp with wavelength 589nm is diffracted by the fabric of an umbrella with a spacing of 0.04mm. If the sodium lamp is 20cm across and 100m from the umbrella, by how many multiples of the lamp width will the diffraction order, viewed through the fabric of the umbrella, appear to be displaced.

Homework Equations

Theta = arcsin (wavelength/d)

where d is the line spacing, 4 x10^5 m in this case

The Attempt at a Solution

It was found that the first order diffraction angle is 0.84 degrees using the equation above.
I have the answer and working here but don't understand what the question is asking:

The lamp subtends (0.2 x 180)/(100 x pi) = 0.11 degrees

Diffracted order subtends 0.84/0.11 = 7.4 lamp widths.

If anyone can explain what is going on here it would be greatly appreciated.

 

[anathema]

Sure, I can help explain the solution to this problem. So, in this problem, you are dealing with diffraction, which is the bending of waves around obstacles or through openings. In this case, the sodium lamp is emitting light with a specific wavelength (589nm) and this light is being diffracted by the fabric of the umbrella, which has a specific spacing (0.04mm).

The first step in solving this problem is to use the diffraction equation, which is θ = arcsin (λ/d), where θ is the diffraction angle, λ is the wavelength of the light, and d is the spacing of the fabric. Plugging in the values given in the problem, we get θ = arcsin (589nm/0.04mm) = 0.84 degrees. This is the angle at which the light is diffracted by the fabric of the umbrella.

Now, the question is asking how many multiples of the lamp width will the diffraction order appear to be displaced when viewed through the fabric of the umbrella. To understand this, we need to think about the geometry of the situation. The lamp is 20cm across and is located 100m away from the umbrella. This means that the lamp subtends an angle of (0.2 x 180)/(100 x pi) = 0.11 degrees when viewed from the umbrella. This is the angle that the lamp appears to take up when viewed through the fabric of the umbrella.

Now, we can compare this to the diffraction angle we calculated earlier (0.84 degrees). We can see that the diffraction angle is much larger than the angle subtended by the lamp. This means that the diffraction pattern will appear much larger than the actual size of the lamp when viewed through the fabric of the umbrella. To find out how many multiples of the lamp width this diffraction pattern will appear to take up, we simply divide the diffraction angle by the angle subtended by the lamp. This gives us 0.84/0.11 = 7.4 lamp widths. This means that the diffraction pattern will appear to take up 7.4 times the width of the lamp when viewed through the fabric of the umbrella.

I hope this explanation helps! Let me know if you have any other questions.

 

[ogb p]

The question is asking how many multiples of the lamp width will the diffraction order appear to be displaced when viewed through the fabric of the umbrella. In order to calculate this, we first need to find the diffraction angle using the equation theta = arcsin (wavelength/d). In this case, the wavelength is 589nm and the line spacing is 0.04mm (or 4 x 10^-5 m). Plugging these values into the equation, we get theta = arcsin (589 x 10^-9 / 4 x 10^-5) = 0.84 degrees.

Next, we need to find the angle subtended by the lamp. This can be done using the formula (object size x 180)/(distance x pi). In this case, the object size is 0.2m (the diameter of the lamp) and the distance is 100m. Plugging these values into the equation, we get 0.11 degrees.

Now, to find the displacement of the diffraction order, we divide the diffraction angle by the angle subtended by the lamp. This gives us 0.84/0.11 = 7.4. This means that the diffraction order will appear to be displaced by 7.4 multiples of the lamp width when viewed through the fabric of the umbrella. This displacement is due to the interference of light waves passing through the closely spaced lines of the umbrella fabric.