Diffraction grating A has 16 000 lines/cm, and diffraction grating B has 12 000 lines/cm. Light of the same wavelength is sent through both gratings to a screen located a distance L from the gratings. The spacing of the maxima for grating A is 3.46 cm. What is the spacing for the maximum for grating B?
To find the spacing for the maximum (also known as the fringe spacing) for grating B, we can use the formula:
d = 1/n * λ * L
where:
- d is the fringe spacing,
- n is the number of lines per unit length (lines/cm),
- λ is the wavelength of the light, and
- L is the distance from the grating to the screen.
Given the information, we have:
- n (grating A) = 16,000 lines/cm
- n (grating B) = 12,000 lines/cm
- d (grating A) = 3.46 cm
- λ (same for both gratings)
Since the light has the same wavelength, we can assume λ is constant for both gratings. Therefore, we can set up the following equation to find the unknown fringe spacing for grating B:
d (grating B) = (n (grating A) / n (grating B)) * d (grating A)
Substituting the given values:
d (grating B) = (16,000 lines/cm / 12,000 lines/cm) * 3.46 cm
Simplifying:
d (grating B) = (4/3) * 3.46 cm
d (grating B) ≈ 4.61 cm
Therefore, the spacing for the maximum for grating B is approximately 4.61 cm.
To find the spacing for the maximum for grating B, we can use the formula:
d * sin(θ) = m * λ
Where:
d is the spacing between the lines on the grating
θ is the angle at which the maximum is observed
m is the order of the maximum
λ is the wavelength of light
Since both gratings are illuminated with light of the same wavelength, λ is constant for both gratings.
Let's assume that for grating A, the maximum is observed at θA, and for grating B, the maximum is observed at θB.
For grating A:
dA * sin(θA) = mA * λ (Equation 1)
For grating B:
dB * sin(θB) = mB * λ (Equation 2)
Since the spacings between the lines on the gratings are different, we need to find a relationship between dA and dB.
Given that grating A has 16,000 lines/cm and a spacing of 3.46 cm:
dA = 1 / 16,000 cm = 0.0000625 cm
Similarly, for grating B with 12,000 lines/cm, we have:
dB = 1 / 12,000 cm = 0.0000833 cm
We can now substitute the values of λ, dA, and dB into Equation 1 and Equation 2:
0.0000625 * sin(θA) = mA * λ (Equation 1)
0.0000833 * sin(θB) = mB * λ (Equation 2)
Since λ is constant for both equations, we can divide Equation 2 by Equation 1 to eliminate λ:
(0.0000833 * sin(θB)) / (0.0000625 * sin(θA)) = (mB * λ) / (mA * λ)
Simplifying further:
0.0000833 / 0.0000625 = mB / mA
1.328 = mB / mA
Since the orders mB and mA are integers, we can conclude that mB must be approximately 1.328 times mA.
So, we can say that mB ≈ 1.328 * mA
Since the spacing for the maximum for grating A is given as 3.46 cm, we substitute mA = 1 and the corresponding value of mB into Equation 2:
0.0000833 * sin(θB) = 1.328 * λ
Now we have sin(θB) in terms of λ:
sin(θB) ≈ (1.328 * λ) / 0.0000833
Therefore, the spacing for the maximum for grating B can be found by using the same formula as grating A but with θB:
dB * sin(θB) = mB * λ
Spacing for the maximum for grating B = dB * sin(θB) = dB * [(1.328 * λ) / 0.0000833]
Substituting the value of dB:
Spacing for the maximum for grating B = 0.0000833 cm * [(1.328 * λ) / 0.0000833]
Simplifying further, we find:
Spacing for the maximum for grating B ≈ 1.328 * λ
Therefore, the spacing for the maximum for grating B is approximately 1.328 times the spacing for grating A.