Laser (Homewoprk 8-323)

4. A laser near Earth beams light in the direction of Earth. It emits red light with intensity $10^{20}$ photons/second (in the laser’s frame). At some initial time, the laser is at rest (relative to the Earth frame).

a. It the initial time, what’s laser power detected at the Earth? Assume the light travel time is small.

b. What’s the recession velocity of the laser (in the Earth’s frame) after 10 years (laser time) elapsed from the initial time? Assume the laser’s rest mass is 10 kg.

c. After those 10 years have elapsed, what’s the laser power detected at Earth? (Ignore the light transit time).

d. Suppose you are the observer on Earth: show that the decreasing power detected over time is consistent with energy conservation.

Initially

$P=N/t hν=10^{20} (hν)$

red light is $hν=1.65-2 eV$ so $P=10^{20}*(1.8*1.6*10^{-19})=28.8 W$

b)

If p is momentum and P is power then

$p/t=P/c$ since $p=E/c$;

non relativistic:

$t=10 yr=10*365*24*3600=3.154*10^8 s$

$F=p/t=P/c = 28.8/(3*10^8 )=9.6*10^{-8} N$ so that $v=at=9.6*10^{-8}*3.154*10^8=30.3 m/s$

c)

$t=γt_0$ so that $P=N/t (hν)=P_0/γ≈P_0$

d)

$p=(P*t)/c$ and $E^2=p^2 c^2+m_0^2 c^4$ or $E^2-E_0^2=p^2 c^2$

$E^2-E_0^2=(P*t)^2$

Therefore decreasing power is consistent with energy conservation.