Question #dc4cc

As you know, frequency and wavelength have an inverse relationship described by the equation

#color(blue)( lamda * nu = c)" "#, where

#lamda# - the frequency of the wave
#nu# - its frequency
#c# - the speed of light in vacuum, usually given as #3 * 10^8"ms"^(-1)#

In simple terms, the product between a wave's wavelength and its frequency will always be equal to the speed of light.

This means that the higher the frequency, the shorter the wavelength, and the lower the frequency, the longer the wavelength.

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Now, as far as the calculations go, the most important thing to make sure that you do correctly is convert the units for the wavelength and for the frequency so that the match the units given for the speed of light.

Notice that the speed of light is expressed in meters per second. This means that you must convert the wavelength to meters and the frequency to Hertz, since

#"1 Hz" = "1 s"^(-1)#

I'll show you how to solve the first part of the question, and leave the second part to you as practice.

So, the wavelength is given to you in nanometers. You must convert this to meters by using the conversion factor

#"1 m" = 10^9"nm"#

Rearrange the equation and solve for #nu#, the frequency of blue light

#lamda * nu = c implies nu = c/(lamda)#

#nu = (3 * 10^8 color(red)(cancel(color(black)("m")))"s"^(-1))/(456 * 10^(-9)color(red)(cancel(color(black)("m")))) = color(green)(6.58 * 10^(14)"Hz"#

If you want, you can express this value in a multiple of Hertz, like gigahertz

#nu = color(green)(6.58 * 10^5"GHz"#

Notice that the short wavelength corresponds to a high frequency, just like described above.

For the second part, convert the frequency from megahertz to Hertz and use the same equation to find the wavelength in meters.

#lamda * nu = c implies lamda = c/(nu)#

If you want, you can then convert the wavelength to something like nanometers.