Physics3, question on intensity, power, wavelength

[Yroyathon]

hi folks, here's another problem I've made a few incorrect guesses on. On my other post I may have not mentioned the potential for errors in using various SI units, but I think in general I'm careful enough so I'll continue to not worry about that in this post. So assume I know enough to convert grams to kilograms, and nanometers to meters, etc.

Homework Statement

A scientist wishes to study the behavior of individual photons. To do that, she must decrease the intensity of her i_1 mm2 laser beam -- the laser emits radiation with wavelength w_1 nm -- to a level at which there is no more than one photon in her apparatus at any given time. The path length of the light beam from source to detector is L_1 m. What should be the intensity?

Homework Equations

for intensity I've used I=P/A, power over area.
for Power of the laser in terms of the photons, I used P = N * E, where N is the number of photons and E is the energy of a photon. for calculating the energy of a photon I used E = h*c/w_1, where h is Planck's constant, c is the speed of light, and w_1 is the wavelength. so in this problem, P = N*6.6*10^(-34)*(3*10^8))/(w_1*10^(-9)).

The Attempt at a Solution

Hmm. I guess I just realized I wasn't sure what to use for A in the equation for intensity. I think in my first guess I just set A=1, without really thinking.

In my guess I realized I didn't use the distance L_1, which I feel is bad. I wasn't really sure how to incorporate it into the problem.

So I'm considering the quantity of L_1/c, which might be how long it takes a photon to reach the detector? but I'm not certain of this.

I'm not getting this right because using my conceptual understanding of the scenario and the equations i have at hand, I don't have a complete cohesive picture.

Any tips would be appreciated. Thanks.

,Yroyathon

 

[Redbelly98]

I'm guessing that "in her apparatus" means in the L₁ m between the source and the apparatus.

So basically, the beam must contain 1 photon per L₁ m of beam length. Can you use that information to say how many photons per second the beam would have?

 

[Yroyathon]

exactly. I'm thinking the quantity L_1/c is the number of photons per second the beam would have. the units seem to match up too, and intuitively i think this number should be pretty small, as it is.

 

[Yroyathon]

well that did not work out. hmmm. i'll think on it more, trying to understand it conceptually, but as always any suggestions would be great.

 

[Redbelly98]

It should have worked out. From L₁/c for the number of photons per sec, and also the energy per photon, you can get

power=energy/sec​

and from there use the area to get the intensity. Looking at the units will be a good way to check the final answer.

 

[Redbelly98]

Aaaack, hold on a minute!

1/L₁ is the number of photons per m.

c/L₁ is the number of photons per second.

 

[Yroyathon]

(also, I'm going to rename i_1 as A, which was a misnomer in my head since that figure is not an intensity but an area)

I got it! it turns out that I had that one figure upside down. the number of photons per second the beam should have is very large actually (my intuition was that it should be small).

so N=c/L_1 was what worked out.

So with I=P/A, and P=NE, and E=hc/w_1, and N=c/L_1, i get

I=P/A=NE/A=(c/L_1)*(h*c/w_1)/A=h*c*c/(L_1*w_1*A)

Awesome! Thanks for your help!

 

[Yroyathon]

exactly!