Question #232b6

Your starting point here will be to use the wavelength of the photons to figure out the energy of a single photon.

To do that, you need to use the Planck - Einstein relation, which looks like this

#E = h * c/(lamda)#

Here

• #h# is Planck's constant, equal to #6.626 * 10^(-34)color(white)(.)"J"#
• #c# is the speed of light in a vacuum, usually given as #3 * 10^8color(white)(.)"m s"^(-1)#
• #lamda# is the wavelength of the photon, expressed in meters

Convert your wavelength from nanometers to meters first

#407 color(red)(cancel(color(black)("nm"))) * "1 m"/(10^9color(red)(cancel(color(black)("nm")))) = 4.07 * 10^(-7)color(white)(.)"m"#

then plug in the value into the equation and solve for #E#, the energy of a single photon of the given wavelength.

#E = 6.626 * 10^(-34)"J" color(red)(cancel(color(black)("s"))) * (3 * 10^8 color(red)(cancel(color(black)("m"))) color(red)(cancel(color(black)("s"^(-1)))))/(4.07 * 10^(-7)color(red)(cancel(color(black)("m"))))#

#E = 4.884 * 10^(-19)color(white)(.)"J"#

Now, you know that your laser pulse contains a total of

#4.67 color(red)(cancel(color(black)("mJ"))) * "1 J"/(10^3color(red)(cancel(color(black)("mJ")))) = 4.67 * 10^(-3)color(white)(.)"J"#

of energy, so you can say that you will need a total of

#4.67 * 10^(-3) color(red)(cancel(color(black)("J"))) * "1 photon"/(4.884 * 10^(-19)color(red)(cancel(color(black)("J")))) = color(darkgreen)(ul(color(black)(9.56 * 10^(15)color(white)(.)"photons")))#

of wavelength #"407 nm"# to generate the energy needed for your laser pulse.

The answer is rounded to three sig figs.