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Assume that I have got a bowl half-filled with mercury. In this bowl, I place a convex lens which make one side of the lens fully reflecting. What will happen to the lens, will it behave like a concave mirror and what will be the change in the sign of the focal length??

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    $\begingroup$ It will behave as a concave mirror with a convex lens in front of it, or just as a convex mirror if the other side is reflecting (with a lens behind it). You would have to do ray tracing to see more precise behavior. $\endgroup$ – user47014 Aug 26 at 2:30
  • $\begingroup$ @user47014. You should make your comment an answer as no answers so far address both cases of which side light is incident on. $\endgroup$ – Paul Childs Aug 26 at 3:04
  • $\begingroup$ @user47014 why not write this in answer? $\endgroup$ – Sciencisco Aug 26 at 3:05
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    $\begingroup$ @Paul Childs what do you mean by which side light is incident on? It's obvious that if I silver a side of a lens (as I can only silver it from outside), I would make the light incidence from the other (non-silvered) side only. $\endgroup$ – user233565 Aug 26 at 3:25
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    $\begingroup$ How is it obvious? Silver is reflective on both sides; whether affixed to glass or not. Curved mirrors are typically used with the glass only used as a substrate. $\endgroup$ – Paul Childs Aug 26 at 5:14
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It will behave as a concave mirror with a convex lens in front of it, or just as a convex mirror if the other side is reflecting (with a lens behind it that doesn't really do anything because it's blocked). You would have to do ray tracing to see more precise behavior for the former case.

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    $\begingroup$ that doesn't make sense. How can you silver the first (incident) surface of the convex lens from the inside just in order to make it behave as a convex mirror or to block the lens behind? We can't possibly silver the surface of a lens from the inside! It has to be done from the outside. $\endgroup$ – user233565 Aug 26 at 3:41
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    $\begingroup$ Just cover the lens with something that reflects, then it's just a curved reflecting surface $\endgroup$ – user47014 Aug 26 at 3:45
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    $\begingroup$ Oh come on dude, things don't work like that. Covering the lens with "something that reflects" means covering the lens with another very thin concavo-convex piece of glass of equal refractive index and radius of curvature as our original convex lens. And in that too, there should be a very little difference in the radius of curvatures of the two surfaces of that piece of glass, with the less convex side silvered on it's back. Summing up, you are making additional modifications to the lens, which are not allowed. The question explicitly mentions that we have to silver one of it's sides! $\endgroup$ – user233565 Aug 26 at 4:17
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    $\begingroup$ It just says a side is made reflecting, it doesn't say how. And anyways I think the question is probably about what happens when it's reflected back through the lens, which I answer also $\endgroup$ – user47014 Aug 26 at 4:21
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    $\begingroup$ You don't have to have glass on a surface to make it reflecting. $\endgroup$ – user47014 Aug 26 at 4:25
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As others have pointed out, it will behave as a lens and a mirror. I wanted to point out a particular version of this with interesting applications: the cat's eye retroreflector:

Cat's eye

The cat's eye retroreflector is a transparent sphere that is silvered on one side, much like the eye of a cat with its reflective tapetum lucidum at the back. This structure has the curious property that it always reflects light directly back at the source. This makes it "light up" if you hit it with a flash light or headlight. This of course, leads to its use on roads

Cat's Eye road reflector

Not all lenses mirrored on one side will have this precise effect, but I think it's pretty awesome!

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Yes. Convex mirrors are typically made by depositing a metal coating on the curved surface of a convex piece of glass. But the focal lengths are incomparable.

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As other answers have pointed out, if one surface of the lens is made reflective by for instance coating it with a thin reflective film, for light incident from the coated side it will behave as a convex mirror, and for light incident from the other side it will behave as a concave mirror. Assuming the lens is sufficiently thin, we can also calculate how "concave" this mirror is, i.e. what the effective focal length is.

Let $R_1$ be the radius of curvature of the non-coated surface and $R_2$ that of the coated surface, both assumed positive if the surface is convex. For an object placed a distance $x$ away from the non-coated side of the system, by the lensmaker's equation we have for the image through the lens at $x'$ $$ \frac{1}{x} + \frac{1}{x'} = (n - 1)\left(\frac{1}{R_1} + \frac{1}{R_2}\right). $$

For the image through the mirror located at $y'$, we have $$ -\frac{1}{x'} + \frac{1}{y'} = \frac{2}{R_2}. $$

For the final image back though the lens located at $y$, we have $$ -\frac{1}{y'} + \frac{1}{y} = (n - 1)\left(\frac{1}{R_1} + \frac{1}{R_2}\right). $$

Adding up all three equations, $$ \frac{1}{f} = \frac{2}{R} = \frac{1}{x} + \frac{1}{y} = 2\frac{n-1}{R_1}+\frac{2}{R_2} $$ where $f$ is the effective focal length and $R$ is the radius of curvature of the equivalent concave mirror. Rearranging, we have

$$ f = \frac{1}{2} \frac{R_1 R_2}{(n-1)R_2 + R_1} $$ $$ R = 2f = \frac{R_1 R_2}{(n-1)R_2 + R_1}. $$

Edit: My above statement that the lens behaves as a concave mirror for light incident from the non-coated side assumes that the lens is biconvex. For a concavo-convex lens, depending on the radii of curvature and the refractive index of the lens, it can also behave as a planar or a convex mirror. If for instance $R_1 < 0$, $R_2 > 0$ and $R_2 < -R_1 < (n-1)R_2$, the lens by itself is still converging but the lens-mirror system behaves as a convex mirror.

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    $\begingroup$ Did you consider a concavo-convex lens? Cause for a biconvex lens, both the radius of curvatures cannot be positive. The one at which light is incident upon, will have a positive roc, and the other surface will have a negative roc, by convention. $\endgroup$ – user233565 Aug 26 at 9:06
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    $\begingroup$ Here I'm assuming $R_1$ and $R_2$ are both positive if the corresponding lens surfaces are convex for light incident from outside the lens. This is not the usual convention used for instance in the lensmaker's equation, but using this convention would have complicated things a little in this case since the light goes through the lens in both directions. I did have the biconvex case in mind, but the equations should hold for a concavo-convex lens as well, by changing the sign of the radius of curvature corresponding to the concave surface. $\endgroup$ – Puk Aug 26 at 9:25

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