A diffraction grating produces a first-order maximum at an angle of 20.0°. What is the angle of the second-order maximum?

Verified step by step guidance

1. Step 1: Recall the diffraction grating equation: $d\; sin(\theta )\; =\; m\lambda$, where $d$ is the distance between adjacent slits (grating spacing), $\theta$ is the diffraction angle, $m$ is the order of the maximum, and $\lambda$ is the wavelength of light.
2. Step 2: For the first-order maximum ($m\; =\; 1$), the angle is given as 20.0°. Use this information to determine the ratio $\lambda /d$ by rearranging the equation: $\lambda /d\; =\; sin(20.0°)$.
3. Step 3: For the second-order maximum ($m\; =\; 2$), substitute $m\; =\; 2$ and the previously calculated $\lambda /d$ into the diffraction grating equation: $sin(\theta \_2)\; =\; 2(\lambda /d)$.
4. Step 4: Solve for $\theta \_2$ by taking the inverse sine: $\theta \_2\; =\; sin-1(2(\lambda /d))$. Use the value of $\lambda /d$ from Step 2.
5. Step 5: Verify that the calculated angle $\theta \_2$ is physically valid (i.e., $sin(\theta \_2)$ must be ≤ 1). If $sin(\theta \_2)$ exceeds 1, the second-order maximum does not exist.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its constituent wavelengths. It works on the principle of interference, where light waves overlap and combine, producing distinct patterns of maxima and minima. The spacing between the grating lines determines the angles at which these patterns occur.

Order of Maximum

The order of maximum refers to the specific bright spots produced in a diffraction pattern, labeled by integers (0, 1, 2, etc.). The first-order maximum is the first bright spot on either side of the central maximum (zeroth order). Each order corresponds to a different angle of diffraction, which can be calculated using the grating equation.

Grating Equation

The grating equation relates the angle of diffraction to the wavelength of light and the spacing of the grating lines. It is expressed as d sin(θ) = nλ, where d is the distance between grating lines, θ is the angle of the maximum, n is the order of the maximum, and λ is the wavelength. This equation allows for the calculation of angles for different orders of maxima.