Show that a diffraction grating cannot produce a second-order maximum for a given wavelength of light unless the first-order maximum is at an angle less than \(30.0^{\circ}\)

The condition for the maxima in a diffraction grating is given by: \(n\lambda = d\sin{\theta}\) Here, \(n\) is the order of the maximum, \(\lambda\) is the wavelength of the light, \(d\) is the distance between the lines of the grating, and \(\theta\) is the angle of the maximum with respect to the straight-through direction.

To find the maximum possible angle for the first-order maximum (n=1), we can set the right side of the maxima condition equation to its maximum value. Since the maximum value of the sine function is 1, we have: \(1\lambda = d\sin{\theta_1}\) To find \(\theta_1\), we can solve for \(\sin\theta_1\): \(\sin\theta_1 = \frac{\lambda}{d}\) Since \(\sin\theta_1 \le 1\), we have \(\frac{\lambda}{d} \le 1\) as the condition for the first-order maximum.

Now we want to find the condition for the second-order maximum (n=2). Using the same formula, we have: \(2\lambda = d\sin{\theta_2}\) We can now solve for \(\sin\theta_2\): \(\sin\theta_2 = \frac{2\lambda}{d}\)

Now, we want to show that the second-order maximum can only occur if the angle of the first-order maximum is less than 30.0°. Let's compare the conditions we found for the first-order and second-order maximum: First-order maximum: \(\sin\theta_1 = \frac{\lambda}{d}\) Second-order maximum: \(\sin\theta_2 = \frac{2\lambda}{d}\) Clearly, \(\sin\theta_1 = \frac{1}{2}\sin\theta_2\). Hence, the angle of the first-order maximum must be less than half the angle of the second-order maximum. Let's now find the maximum possible value of \(\theta_1\). Since \(\sin\theta_1 \le 1\), we have: \(\theta_1 \le \arcsin{1} = 90^{\circ}\) Therefore, the maximum possible value of the angle for the second-order maximum is twice the angle of the first-order maximum; hence, \(\theta_2 \le 2\times 90^{\circ} = 180^{\circ}\). Now, if we take the second-order maximum (n=2) to be \(60^{\circ}\), then the first-order maximum will be half this value, which is exactly 30.0\(\,^{\circ}\).

Since the angle of the first-order maximum must be less than half the angle of the second-order maximum, it must be less than 30.0\(\,^{\circ}\) when a second-order maximum exists. Therefore, we have shown that a diffraction grating cannot produce a second-order maximum for a given wavelength of light unless the first-order maximum is at an angle less than 30.0\(\,^{\circ}\).