Light of wavelength 589 nm in vacuum passes through a piece of fused quartz of index of refraction n = 1.458…

Analysis

To solve this problem, we need to understand the relationship between the speed of light, frequency, wavelength, and the refractive index. A key principle in physics is that the frequency of a light wave does not change when it passes from one medium to another; only its speed and wavelength change.

Concept

The fundamental wave equation is:

Where:

• $c$ = speed of light in vacuum ($3.00 \times 10^8 \text{ m/s}$)
• $f$ = frequency (in Hertz)
• $\lambda$ = wavelength (in meters)

Since the frequency ($f$) remains constant regardless of the medium, we can calculate it using the values provided for the vacuum.

Step-by-Step Solution

1. Identify the given values:

• Wavelength ($\lambda$) = $589 \text{ nm} = 589 \times 10^{-9} \text{ m}$
• Speed of light ($c$) = $3.00 \times 10^8 \text{ m/s}$
• Refractive index ($n$) = $1.458$ (Note: This is extra information provided to test if you know that frequency remains constant!)

2. Rearrange the formula to solve for frequency ($f$):

3. Substitute the values:

4. Perform the calculation:

To express this in standard scientific notation:

The refractive index ($n = 1.458$) is irrelevant to the frequency calculation because the frequency is determined by the source of the light, not the medium it travels through.

Conclusion

Comparing our result to the given options, the correct value is $5.09 \times 10^{14} \text{ Hz}$. This matches option B.

Exam Tip

Always remember the "Frequency Rule": When light travels between different media (like air to glass or water), frequency stays the same, but speed and wavelength change. If a question gives you the refractive index but asks for frequency, don't let it distract you—the frequency is the same as it was in the vacuum!