Oil immersion lens
 

An insect is embedded in a glass sphere with radius R
and refractive index mu.
It is R/mu from the centre, and is emitting rays obeying Snell's Law.

Show that half of the rays emerge from a point R*mu from the centre.

David

PS I'm sure ccorn can produce a beautiful picture
of this.

[QUOTE=davieddy;219157]It is R/mu from the centre, and is emitting rays obeying Snell's Law.[/QUOTE]When I took math for laser optics years ago I could have done this one.

Dynamite with a laser beam
 

[quote=Uncwilly;219158]When I took math for laser optics years ago I could have done this one.[/quote]
[URL]http://www.youtube.com/watch?v=z_4Jf18XDoo&feature=fvst[/URL]

[QUOTE=davieddy;219157]An insect is embedded in a glass sphere with radius R and refractive index mu.
It is R/mu from the centre, and is emitting rays obeying Snell's Law.

Show that half of the rays emerge from a point R*mu from the centre.
[...]
PS I'm sure ccorn can produce a beautiful picture of this.[/QUOTE]
And here it is. [strike]The subscript "oil" actually means the effects of glass and oil combined.[/strike]

Edit: Apart from an infinitesimal parallel shift, the glass of the sphere should not influence the light path.

For those who have Mathematica or the free Mathematica Reader...

OK, here's a version that will actually work with the new free [URL="http://www.wolfram.com/products/player/download.cgi"]Mathematica Player[/URL]. Either version will work with the full Mathematica software.

[quote=ccorn;219225]And here it is. [strike]The subscript "oil" actually means the effects of glass and oil combined.[/strike]

Edit: Apart from an infinitesimal parallel shift, the glass of the sphere should not influence the light path.[/quote]
Hmm.
Could do better.
The sphere is solid glass.
The relevance of the oil will be explained in due course.

David

[QUOTE=davieddy;219251]The sphere is solid glass.
The relevance of the oil will be explained in due course.[/QUOTE]
Then I suppose you mean a glass [I]ball[/I] and not a glass [I]sphere[/I].
Updated figure attached. I have also changed the value of [tex]\mu[/tex] from 3 to 2 which is more realistic for glass.

[QUOTE=frmky;219246]OK, here's a version that will actually work with the new free [URL="http://www.wolfram.com/products/player/download.cgi"]Mathematica Player[/URL]. Either version will work with the full Mathematica software.[/QUOTE]
Thanks! As another variant, here is a draft for the [url=http://www.geogebra.org/cms/]GeoGebra[/url] system which runs in a browser or as a Java WebStart app.

[QUOTE=davieddy;219157]An insect is embedded in a glass sphere with radius R and refractive index mu.
It is R/mu from the centre, and is emitting rays obeying Snell's Law.

Show that half of the rays emerge from a point R*mu from the centre.[/QUOTE]
And the other half appears as if having emerged from the same outside point, but reflected by the glass [strike]sphere[/strike]ball.

I know ccorn is chomping at the bit, but I haven't demonstrated
this gem since I stopped teaching in 1987.

Refer to ccorn's diagram. Not sure why he has designated the
angles by delta, since they aren't small. I'll call delta glass i
and delta air r.

First of all the triangles COS' and SOC are similar because the
angle at O os common, and S'O/CO = CO/SO = mu (Given).

It follows that angle OCS = angle OS'C and (not relevant here
but crucial for the gravity problem) S'C/CS = mu.

By applying the sine rule to either triangle, we get sin i/sin r = mu
(Snell's Law) which follows from Fermat's Principle (for all you
number theory fans out there:).

But this only works when CSO < 90 degrees (i = critical angle of incidence).

I shall now consider ccorn's last post!

David