Question

Plane microwaves are incident on a single slit of width \(2.00 \mathrm{~cm}\). The second minimum is observed at an angle of \(43.0^{\circ} .\) What is the wavelength of the microwaves? \(?\) ? microwave??

Short answer

Answer: The wavelength of the microwaves is \(\lambda = 6.82\times10^{-3}\;m\).

Step-by-step solution

Identify the formula for single-slit diffraction

We will use the formula for single-slit diffraction to find the wavelength of the microwaves. The general formula for the minima is: \(Sin(\theta_n) = \frac{n\lambda}{a}\) where \(n\) is the index of the minimas, \(a\) is the width of the slit, \(\lambda\) is the wavelength, and \(\theta_n\) is the angular position of the nth minimum. Since we need to find the wavelength of the microwaves, we will solve the equation for \(\lambda\).

Plug in the given values and solve for the wavelength

We are given the width of the slit \(a=2.00\;cm = 0.02\;m\), and the angle at which the second minimum is observed, \(\theta=43.0^{\circ}\). The index \(n\) for the second minimum is \(n=2\). First, we need to convert the given angle from degrees to radians: \(\theta_{rad} = \frac{43.0^{\circ} * \pi}{180^\circ} = 0.750\;rad\) Now we can plug these values into the single-slit diffraction formula: \(Sin(0.750) = \frac{2\lambda}{0.02}\) Solve for the wavelength: \(\lambda = \frac{0.02*Sin(0.750)}{2} = \frac{0.02*0.682}{2} = 6.82\times10^{-3}\;m\)

Present the solution

The wavelength of the microwaves is \(\lambda = 6.82\times10^{-3}\;m\).