THE DROWNING CHILD, LIFEGUARD AND SNELL’S LAW

Consider a rectangular swimming pool PQSR; see figure here. A lifeguard sitting at G outside the pool notices a child drowning at a point C. The guard wants to reach the child in the shortest possible time. Let SR be the side of the pool between G and C.

Should he/she take a straight line path GAC between G and C or GBC in which the path BC in water would be the shortest, or some other path GXC? The guard knows that his/her running speed v1 on ground is higher than his/her swimming speed v2. Suppose the guard enters water at X. Let GX =l1 and XC =l .

Then the time taken to reach from G to C would be l l t v v To make this time minimum, one has to differentiate it (with respect to the coordinate of X ) and find the point X when t is a minimum. On doing all this algebra (which we skip here), we find that the guard should enter water at a point where Snell’s law is satisfied. To understand this, draw a perpendicular LM to side SR at X. Let GXM = i and CXL = r.

Then it can be seen that t is minimum when sin sin v i r v In the case of light v1/v2, the ratio of the velocity of light in vacuum to that in the medium, is the refractive index n of the medium. In short, whether it is a wave or a particle or a human being, whenever two mediums and two velocities are involved, one must follow Snell’s law if one wants to take the shortest time. This is called total internal reflection. When light gets reflected by a surface, normally some fraction of it gets transmitted.

The reflected ray, therefore, is always less intense than the incident ray, howsoever smooth the reflecting surface may be. In total internal reflection, on the other hand, no transmission of light takes place. The angle of incidence corresponding to an angle of refraction 90º, say AO3N, is called the critical angle (ic ) for the given pair of media. We see from Snell’s law [Eq.

( . )] that if the relative refractive index is less than one then, since the maximum value of sin r is unity, there is an upper limit to the value of sin i for which the law can be satisfied, that is, i = ic such that sin ic = n ( . ) For values of i larger than ic, Snell’s law of refraction cannot be satisfied, and hence no refraction is possible. The refractive index of denser medium with respect to rarer medium will be n12 = /sinic.

Some typical critical angles are listed in Table . . FIGURE . Refraction and internal reflection of rays from a point A in the denser medium (water) incident at different angles at the interface with a rarer medium (air).

A demonstration for total internal reflection All optical phenomena can be demonstrated very easily with the use of a laser torch or pointer, which is easily available nowadays. Take a glass beaker with clear water in it. Stir the water a few times with a piece of soap, so that it becomes a little turbid. Take a laser pointer and shine its beam through the turbid water.

You will find that the path of the beam inside the water shines brightly. Shine the beam from below the beaker such that it strikes at the upper water surface at the other end. Do you find that it undergoes partial reflection (which is seen as a spot on the table below) and partial refraction [which comes out in the air and is seen as a spot on the roof; Fig. .

(a)]? Now direct the laser beam from one side of the beaker such that it strikes the upper surface of water more obliquely [Fig. . (b)].

Adjust the direction of laser beam until you find the angle for which the refraction