At which wavelength does a 100 mW laser deliver 1.6 × 1017 photons in one second? Let 1 eV = 1.60 × 10−19 J, the speed of light c = 3.00 × 108 m/s, and Planck’s constant h = 4.136 × 10−15 eV ∙ s.

Question

Peyton Adams · Accepted Answer

320 nm

I'll be working in Joules exclusively. The power rating of the laser is 100mW. This is #sf(100xx10^(-3)=0.1color(white)(x)"J/s")# This means that #sf(1.6xx10^(17))# photons deliver 0.1 J of energy every second. #:.# the energy of 1 photon = #sf(0.1/(1.6xx10^(17))=0.0625xx10^(-17)color(white)(x)J)# This allows us to apply the Planck Expression: #sf(E=hf=(hc)/lambda)# #:.# #sf(lambda=(hc)/E)# The SI value of h is #sf(6.63xx10^(-34)color(white)(x)J.s)# #sf(lambda=(6.63xx10^(-34)xx3.00xx10^8)/(0.0625xx10^(-17))color(white)(x)m)# #sf(lambda=318.2xx10^(-9)color(white)(x)m)# #sf(lambda=320color(white)(x)nm)#

Peyton Adams · Answer

undefined The wavelength can be calculated using the formula: $$\lambda = \frac{hc}{E}$$
where:
- $h$ is Planck's constant (given as $4.136 \times 10^{-15}$ eV · s)
- $c$ is the speed of light ($3.00 \times 10^8$ m/s)
- $E$ is the energy of one photon, which can be calculated from the given power of the laser (100 mW) and the number of photons (1.6 × $10^{17}$)
- Convert energy from Joules to electron volts (eV) using the given conversion factor (1 eV = $1.60 \times 10^{-19}$ J)

After calculating $E$, plug it into the formula to find $\lambda$.