Problem P. 5351. (October 2021)
P. 5351. Why is it not allowed to look into a laser light?
The lens in the human eye can focus the light rays into a very small spot of diameter of several \(\displaystyle \mu\)m. The most sensitive cells are in the retina, and here the size of the cells so called rods and cones is in the range of \(\displaystyle \mu\)m. The power of lasers used in everyday life is between 0.1 mW and 100 mW.
Calculate how long it takes for the light of the smallest laser, i.e. 0.1 mW, with 80% light absorption, to heat a cell to \(\displaystyle 50\,{}^\circ\)C at which the cell gets damaged, and how long it takes to heat the cell to a temperature of \(\displaystyle 100\,{}^\circ\)C at which the cell is totally destroyed. For the sake of simplicity consider the cell as a cylinder which has a base diameter of \(\displaystyle 5~\mu\)m and a height of \(\displaystyle 7~\mu\)m; the density and the specific heat capacity of the cell can be assumed to be the same as those of water. The temperature of the eye can be considered as \(\displaystyle 36\,{}^\circ\)C, and other effects (as displacements, heat conduction etc.) can be neglected. Compare the gained time with the approximately 0.2 second reaction time of the human eye.
(4 pont)
Deadline expired on November 15, 2021.
Sorry, the solution is available only in Hungarian. Google translation
Megoldás. Az idegsejt megadott mérete esetén a térfogata
\(\displaystyle V=\frac{d^2\pi}{4}\cdot h=\frac{\left(5\cdot10^{-6}~ {\rm m}\right)^2\pi}{4}\cdot \left(7\cdot10^{-6}~ {\rm m}\right)= 1{,}4\cdot 10^{-16}~\rm m^3, \)
a tömege
\(\displaystyle m=V\,\varrho_\text{víz}=1{,}4\cdot 10^{-13}~\rm kg.\)
Ennyi víz 1 fokkal való melegítéséhez
\(\displaystyle Q=c_\text{víz}m\Delta T=4{,}2~\frac{\rm kJ}{\rm kg\,K}\cdot (1{,}4\cdot 10^{-13}~\rm kg)\cdot(1~\rm K) \approx 6\cdot 10^{-10} \rm J\)
energia (hő) szükséges. A legkisebb teljesítményű lézer ennyi energiát 80%-os hatásfokkal
\(\displaystyle t_0=\frac{6\cdot 10^{-10} \rm J}{0{,}08~ \rm mW}=7{,}5~\mu\rm s\)
alatt tud leadni.
A fókuszált lézerfény tehát \(\displaystyle 14\,t_0=0{,}1\) ms alatt melegíti az idegsejtet a már károsodást okozó \(\displaystyle 50~^\circ\)C-ra, a \(\displaystyle 100~^\circ\)C-ra történő felmelegítéshez pedig kb. 0,5 ms szükséges. Mindkettő lényegesen rövidebb idő, mint az emberi szem 200 ms-os reakcióideje, tehát a lézerfény okozta szemkárosodást csak úgy tudjuk elkerülni, hogy nem nézünk a lézerfénybe.
Statistics:
77 students sent a solution. 4 points: Bagu Bálint, Barna Benedek, Beke Botond, Gábriel Tamás, Hauber Henrik, Jeszenői Sára, Kertész Balázs, Klepáček László, Kovács Kinga, Mészáros Ádám, Murai Dóra Eszter, Schmercz Blanka, Szabó Márton. 3 points: Albert Máté, Antalóczy Szabolcs, Bacsó Dániel, Bánovics Anna, Bányai Kristóf, Bencz Benedek, Bogdán Benedek, Bubics Gergely Dániel, Csonka Illés, Czirók Tamás, Dobre Zsombor, Kiss-Beck Regina, Kovács Kristóf , Kürti Gergely, Marozsi Lenke Sára, Molnár Kristóf, Molnár-Szabó Vilmos, Nagy 456 Imre, Szanyi Attila, Tatár Ágoston, Vágó Botond, Veszprémi Rebeka Barbara, Waldhauser Miklós, Yokota Adan. 2 points: 19 students. 1 point: 6 students. 0 point: 2 students. Unfair, not evaluated: 9 solutionss.
Problems in Physics of KöMaL, October 2021