Finding refractive index of liquid with no parallax principle

[MadmanMurray]

In physics class we did an experiment where we dropped a metal pin in a beaker of water, covered half the beaker with a mirror and clamped another metal pin above the beaker. We had to find the refractive index of the water by measuring the apparent depth of the submerged pin and dividing it into the real depth of the pin.

The teacher told us we can find out the apparent depth by adjusting the height of the clamped pin so that when looking down the beaker we should be able to see the reflection (in the mirror) of the clamped pin aligned with the visible submerged pin and if we move our heads from side to side there should be no parallax.

It was dead simple to get the right results but I have absolutely no idea how it worked. I know that light rays travel through water faster than they do air and that the light rays deflect a little when they enter the water but I have no idea how this principle can be used to measure the apparent depth of a submerged object. The teacher didn't explain it at all so I don't think we have to know this but I'd really like to know. I've been pondering it trying to come up with an explanation but I can't think of a theory that makes sense.

I should mention that the apparent image was measured as the distance from the clamped pin to the mirror/top of beaker and obviously the real depth was the distance from the submerged pin to the mirror. How come when you adjust the height of the pin on the clamp to the right height above the beaker its reflection in the mirror can be aligned with the visible pin with no parallax. If you raise the pin a little there will be parallax though. Can anyone explain this to me?

 

[Doc Al]

It was dead simple to get the right results but I have absolutely no idea how it worked. I know that light rays travel through water faster than they do air and that the light rays deflect a little when they enter the water but I have no idea how this principle can be used to measure the apparent depth of a submerged object. The teacher didn't explain it at all so I don't think we have to know this but I'd really like to know. I've been pondering it trying to come up with an explanation but I can't think of a theory that makes sense.

Imagine that the submerged pin is a source of light (reflected light, but that doesn't matter). Treating the pin as a point source, draw a diagram showing light "rays" emanating in all directions, being refracted at the air/water interface. By "tracing back" the light rays in air, you can see the apparent position of the pin. Use Snell's law--and a bit of trig--to calculate the apparent depth for rays close to normal incidence.