Visible light is emitted from the sun with a frequency of \( 4,5 \times 10^{13} \mathrm{~Hz} \). Calculate the wavelength of the visible light in \( \mu \mathrm{m} \), rounded to two decimals

Calculate or simplify the expression \( \frac{3e^{2}}{4\pi\epsilon_{0}c^{3}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{3e^{2}}{4\pi \epsilon_{0}c^{3}}\) The value of the constant \( \frac{3e^{2}}{4\pi\epsilon_{0}c^{3}} \) is approximately \( 8.9875 \times 10^{9} \) Coulomb squared per kilogram meter cubed per second squared. Now, we can use this value to calculate the wavelength of the visible light emitted from the sun with a frequency of \( 4.5 \times 10^{13} \) Hz. The formula to calculate the wavelength of light is given by: \[ \lambda = \frac{c}{f} \] where: - \( \lambda \) is the wavelength of the light - \( c \) is the speed of light in vacuum (\( 3 \times 10^{8} \) m/s) - \( f \) is the frequency of the light Substitute the values into the formula to find the wavelength of the visible light in meters: \[ \lambda = \frac{3 \times 10^{8}}{4.5 \times 10^{13}} \] Now, we can calculate the wavelength in meters and then convert it to micrometers (\( \mu \)m) by multiplying by \( 10^{-6} \) to get the final answer. Calculate the value by following steps: - step0: Calculate: \(\frac{3\times 10^{8}}{4.5\times 10^{13}}\) - step1: Rewrite the expression: \(\frac{3\times 10^{8}}{45\times 10^{12}}\) - step2: Reduce the fraction: \(\frac{10^{8}}{15\times 10^{12}}\) - step3: Rewrite the expression: \(\frac{5^{8}\times 2^{8}}{15\times 10^{12}}\) - step4: Rewrite the expression: \(\frac{5^{8}\times 2^{8}}{5\times 3\times 10^{12}}\) - step5: Reduce the fraction: \(\frac{5^{7}\times 2^{8}}{3\times 10^{12}}\) - step6: Rewrite the expression: \(\frac{5^{7}\times 2^{8}}{3\times 5^{12}\times 2^{12}}\) - step7: Reduce the fraction: \(\frac{1}{3\times 5^{5}\times 2^{4}}\) - step8: Calculate: \(\frac{1}{150000}\) The wavelength of the visible light emitted from the sun with a frequency of \( 4.5 \times 10^{13} \) Hz is approximately \( 6.666667 \times 10^{-6} \) meters. To convert this to micrometers (\( \mu \)m), we multiply by \( 10^{-6} \): \[ 6.666667 \times 10^{-6} \times 10^{-6} = 6.666667 \times 10^{-12} \, \mu \text{m} \] Therefore, the wavelength of the visible light in micrometers is approximately \( 6.666667 \times 10^{-12} \, \mu \text{m} \), rounded to two decimals.