While studying reflection through a plane mirror, I have been told that when the object is real the image will be virtual and the image will be real while the object is virtual.
What are virtual objects? Do they really exist in nature?
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Rays (in geometric optics) actually pass through real images, and real objects.
No rays pass through either virtual objects or virtual images; it just appears the rays come from them (or go to them).
To expand on this a bit; consider a simple biconvex lens being used to form an image of an object placed on the axis, at some distance from the lens; somewhere around twice the focal length of the lens. (doesn't matter much where).
The first surface of the lens (traditionally the left surface for left to right propagation), is a boundary between say air, and the lens medium of refractive index N .
If the lens medium was thick (very thick), the first (spherical) surface, will converge the rays from the object, and form a (REAL) image, in the medium of index N.
But in our actual biconvex lens, the thickness, is actually quite small (compared to very thick), so long before the rays can get to that real image point, they encounter the second surface of the lens, which is a boundary between the medium of index N, and the air.
But the rays inside the real lens, are still travelling along the same paths that they were in our imaginary very thick medium, and we know they were heading for some point which is way past the second surface of the real lens.
That point, is now a VIRTUAL IMAGE, (in the medium of index N), that the rays were headed towards, but now will never make it. This VIRTUAL IMAGE (in the medium of index N ) of the first surface refraction, now becomes a VIRTUAL OBJECT (in the medium of index N), for the second surface of our real lens, and the second surface will form a REAL IMAGE (in air) of the VIRTUAL OBJECT (in the medium of index N).
So even in the simplest, or most complex, imaging lens (or mirror) systems, there can be both real and virtual objects and images all over the place.
In our first situation of the very thick medium, we saw, how a single refraction can form a real image of a real object, each being in a different medium. In the second real lens case, we still have objects and images in different media (on each side of a surface), but now one of the pair is real and the other is virtual.
A SINGLE refraction or reflection can form either REAL or VIRTUAL images of REAL or VIRTUAL objects; image and object always being in different media.
We normally treat reflection as a special case of refraction, with the input side index being (1) and the output side index being (-1). Or they could be +N and -N respectively (or in reverse order).
So virtual objects and images are very common; so in that sense we could say they are quite "real".
As has been discussed in the comments, virtual object and image is one or the same thing. The difference in terminology is that suppose you stand in front of mirror you see your virtual image in the mirror, this image can not be obtained on a screen, but now if you use another plane mirror or other lens/mirror combination your image(virtual) in the mirror will act as a virtual object for the other mirror, the image that the new mirror will form will be real and can be obtained on a screen.
No, these do not exist in nature(freely) as these are not even actually formed, you know the rays do not actually meet there, it is kind of like an illusion, we sense the rays to be coming from a point behind the mirror while in reality these rays come from a real object only after reflection. By not existing freely i mean these come into play whenever there is a lens/mirror combination but without them they do not exist.