Question

Question #232b6

Answer 1

`9.56 * 10^(15)`

Explanation:

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.