Luminous Efficacy Concept

Luminous efficacy quantifies how effectively a light source converts power into visible light. It is measured in lumens per watt (lm/W). The perceived brightness, and thus luminous efficacy, is not uniform across all colors (wavelengths) because the human eye has different sensitivities.

Calculating Wavelength from Frequency

The question provides the frequency of the monochromatic radiation and asks for its luminous efficacy in lm/W. First, we need to determine the wavelength ($\lambda$) corresponding to the given frequency ($f$). The relationship between the speed of light ($c$), wavelength, and frequency is:

$ c = \lambda f $

We are given:

• Frequency, $f = 540 \times 10^{12}$ Hz (which is 540 THz)
• The speed of light, $c \approx 3 \times 10^8$ m/s

We can rearrange the formula to solve for wavelength:

$ \lambda = \frac{c}{f} $

Substitute the values:

$ \lambda = \frac{3 \times 10^8 \text{ m/s}}{540 \times 10^{12} \text{ Hz}} $

Calculate the result:

$ \lambda = \frac{3}{540} \times 10^{(8-12)} \text{ m} = \frac{1}{180} \times 10^{-4} \text{ m} $

To convert this wavelength to nanometers (nm), we use the conversion factor $1 \text{ m} = 10^9 \text{ nm}$:

$ \lambda = \left(\frac{1}{180} \times 10^{-4}\right) \times 10^9 \text{ nm} = \frac{10^5}{180} \text{ nm} $

$ \lambda = \frac{100000}{180} \text{ nm} \approx 555.56 \text{ nm} $

Eye Sensitivity and Maximum Luminous Efficacy

The sensitivity of the human visual system peaks in the green region of the visible spectrum, around 555 nm. The standardized luminosity function, $V(\lambda)$, represents this sensitivity, where $V(\lambda) = 1$ at the peak (555 nm) and decreases towards zero at longer and shorter wavelengths.

The luminous efficacy ($K(\lambda)$) of monochromatic light at a specific wavelength ($\lambda$) is related to the maximum possible efficacy ($K_{max}$) by the formula:

$ K(\lambda) = K_{max} \times V(\lambda) $

The maximum luminous efficacy ($K_{max}$) is defined as 683 lm/W, and it occurs at the wavelength where the eye's sensitivity is highest ($\approx 555$ nm).

Determining the Efficacy for 540 THz Radiation

Our calculation shows that the wavelength corresponding to a frequency of $540 \times 10^{12}$ Hz is approximately $555.56$ nm. This wavelength is very close to the 555 nm peak sensitivity of the human eye.

At this wavelength, the value of the luminosity function $V(\lambda)$ is approximately 1.

Therefore, the luminous efficacy ($K_{cd}$) for this monochromatic radiation is:

$ K_{cd} \approx 683 \text{ lm/W} \times V(555.56 \text{ nm}) \approx 683 \times 1 \text{ lm/W} $

$ K_{cd} \approx 683 \text{ lm/W} $

Thus, the luminous efficacy of monochromatic radiation with a frequency of $540 \times 10^{12}$ Hz is approximately 683 lumens per watt.