Two planes of atoms in a crystal are separated by $3.25 \times 10-10 \mathrm{~m}$. Light of what wavelength would show constructive interference at an angle of 17.5 degrees relative to the crystal surface?
A) $1.95 \times 10-10 \mathrm{~m}$
B) $9.75 \times 10^{-10} \mathrm{~m}$
C) $1.46 \times 10^{-10} \mathrm{~m}$
D) $3.9 \times 10-10 \mathrm{~m}$
E) $4.9 \times 10^{-9} \mathrm{~m}$

Question

Quizplus · Accepted Answer

Constructive interference occurs when the path length difference between two waves is an integer multiple of the wavelength. In this case, the path length difference is $2d \sin \theta$, where $d$ is the distance between the planes of atoms and $\theta$ is the angle of incidence. For constructive interference, we have:$$2d \sin \theta = m\lambda$$where $m$ is an integer. Solving for $\lambda$, we get:$$\lambda = \frac{2d \sin \theta}{m}$$Substituting the given values, we get:$$\lambda = \frac{2(3.25 \times 10^{-10} \mathrm{~m}) \sin 17.5^\circ}{1} = 1.95 \times 10^{-10} \mathrm{~m}$$Therefore, the correct answer is A.

Two Planes of Atoms in a Crystal Are Separated By 3.25