A diffraction grating has 5540lines/cm . When a beam of monochromatic light goes through it, the second pair of bright spots occurs at ±26.0cm from the central spot on a screen that is 42.5cm past the grating.

To see if the 'small angle approximation' is sufficient, Calculate tan θ using the quantities given. If sin θ is very different from tan θ then the approximation is not going to be useful. (In the context of these questions, very different means different if we calculate three significant figures.)
question 1- What is the wavelength of this light?
question 2- What angle is between a line from the grating to the central spot, and a line from the grating to the third bright spot (m = 3) on the screen?

N=5540 lines/cm=554000 lines/m

d=1/N=1.8•10⁻⁶ m
1)
m=2
tanφ₂= x₂/L=26/42.5 =0.61
φ₂=tan⁻¹0.61= 31.5⁰
sinφ₂=sin 31.5⁰=0.52
dsinφ =mλ
dsinφ₂=2λ
λ= dsinφ₂/m=1.8•10⁻⁶ •0.52/2=4.5•10⁻⁷ m
2)
m=3
dsinφ₃=3λ
sinφ₃=3 λ/d=3 λ/1.8•10⁻⁶=
=0.75,
φ₃=sin⁻¹0.75=48.8⁰

To answer these questions, we'll follow the steps provided and use the given information.

Step 1: Calculate tan θ using the given quantities.
tan θ = (±26.0 cm) / (42.5 cm)

Step 2: Determine if sin θ is very different from tan θ.

To do this, we need to compare sin θ to tan θ. If they are significantly different, then the small angle approximation will not be useful.

Step 3: Calculate the wavelength of the light.

We'll use the formula for the grating equation:
m * λ = d * sin θ

Here, m refers to the order of the bright spot (for example, m = 1 for the first bright spot, m = 2 for the second bright spot, and so on), λ is the wavelength of light, d is the spacing between the lines on the diffraction grating, and θ is the angle between the line from the grating to the central spot and the line from the grating to the bright spot.

Step 4: Calculate the angle between the line from the grating to the central spot and the line from the grating to the third bright spot (m = 3).

Similar to step 3, we'll use the grating equation with m = 3 to calculate the angle.

Now, let's proceed to solve these questions.

Question 1: What is the wavelength of this light?

To calculate the wavelength of light, we need to find the angle θ.

tan θ = (±26.0 cm) / (42.5 cm)

Next, calculate sin θ by dividing tan θ by √(1 + tan^2 θ). This will give us the value of sin θ.

sin θ = (tan θ) / √(1 + tan^2 θ)

Finally, to find the wavelength, use the grating equation:

m * λ = d * sin θ

Given the values of d (5540 lines/cm) and m (2 for the second bright spot), we can rearrange the equation to solve for λ:

λ = (d * sin θ) / m

Substitute the calculated value of sin θ and the given values of d and m into the equation to find the wavelength.

Question 2: What angle is between a line from the grating to the central spot and a line from the grating to the third bright spot (m = 3) on the screen?

Using the same procedure as in Question 1, calculate the angle θ using the given values (±26.0 cm for the third bright spot and 42.5 cm for the distance from the grating to the screen).

Then use the grating equation with m = 3 to calculate the angle. Rearrange the equation to solve for θ:

θ = arcsin((m * λ) / d)

Substitute the calculated value of λ and the given values of d and m into the equation to find the angle θ.

Follow these steps, perform the calculations, and you'll find the answers to both questions.