Oil immersion lens

[davieddy]

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.

[Uncwilly]

Originally Posted by davieddy

It is R/mu from the centre, and is emitting rays obeying Snell's Law.

When I took math for laser optics years ago I could have done this one.

[davieddy]

Originally Posted by Uncwilly

When I took math for laser optics years ago I could have done this one.

http://www.youtube.com/watch?v=z_4Jf18XDoo&feature=fvst

[ccorn]

Originally Posted by davieddy

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.

And here it is. The subscript "oil" actually means the effects of glass and oil combined.

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

[frmky]

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

[frmky]

OK, here's a version that will actually work with the new free Mathematica Player. Either version will work with the full Mathematica software.

[davieddy]

Originally Posted by ccorn

And here it is. The subscript "oil" actually means the effects of glass and oil combined.

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

Hmm.
Could do better.
The sphere is solid glass.
The relevance of the oil will be explained in due course.

David

[ccorn]

Originally Posted by davieddy

The sphere is solid glass.
The relevance of the oil will be explained in due course.

Then I suppose you mean a glass ball and not a glass sphere.
Updated figure attached. I have also changed the value of from 3 to 2 which is more realistic for glass.

[ccorn]

Originally Posted by frmky

OK, here's a version that will actually work with the new free Mathematica Player. Either version will work with the full Mathematica software.

Thanks! As another variant, here is a draft for the GeoGebra system which runs in a browser or as a Java WebStart app.

[ccorn]

Originally Posted by davieddy

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.

And the other half appears as if having emerged from the same outside point, but reflected by the glass sphereball.

[davieddy]

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