Question

What is the ratio of velocities of light rays of wavelengths $4000^{\circ} \mathrm{A}\( and \)8000^{\circ} \mathrm{A}$ in vacuum? (A) \(1: 2\) (B) \(1: 1\) (C) \(2: 1\) (D) cannot be determined

Short answer

The ratio between the velocities of light rays with given wavelengths is equal to the ratio of their frequencies. We find that the ratio of frequencies for the wavelengths \(4000^{\circ} \mathrm{A}\) and \(8000^{\circ} \mathrm{A}\) is \(2:1\). Therefore, the correct answer is (C) \(2:1\).

Step-by-step solution

Determine the relationship between speed, frequency, and wavelength of light in vacuum.

The relationship between the speed of light (\(v\)), its frequency (\(f\)), and its wavelength (\(\lambda\)) in vacuum is given by the formula: \(v = f\lambda\)

Identify the constant speed of light in vacuum.

Regardless of the wavelength of light - the speed of light (\(c\)) in vacuum is always constant, with the value of: \(c = 3 \times 10^8 \ m/s\)

Calculate the frequency for the given wavelengths.

We want to find the ratio of velocities of light rays with wavelengths \(\lambda_1 = 4000^{\circ} \mathrm{A}\) and \(\lambda_2 = 8000^{\circ} \mathrm{A}\). Using the relationship \(v = f\lambda\), we have: \(f_1 = \frac{c}{\lambda_1}\) and \(f_2 = \frac{c}{\lambda_2}\)

Find the ratio of frequencies.

Now, let's find the ratio of frequencies \(f_1\) and \(f_2\). Here, we have: \(\frac{f_1}{f_2} = \frac{\frac{c}{\lambda_1}}{\frac{c}{\lambda_2}} = \frac{\lambda_2}{\lambda_1} = \frac{8000^{\circ} \mathrm{A}}{4000^{\circ} \mathrm{A}} = \frac{8000}{4000} = 2\)

Confirm the answer using the given options.

The ratio between the velocities of light rays with given wavelengths is equal to the ratio of their frequencies, which we found as \(2:1\). Therefore, the correct answer is (C) \(2:1\).