In an experiment to find focal length of a concave mirror, first we had to estimate its rough focal length. I kept an object at point which was beyond the focus in front of the mirror and got its real image on the screen. But as soon as I removed the screen, I could even see an inverted image of the needle behind the mirror (i.e., virtual image). How is this possible as the concave mirror always form a real image for an object kept at a distance greater than the focal length in front of it?
[Physics] Why does concave mirror simultaneously form two images
experimental-physicsgeometric-opticsopticsreflection
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A real image can be viewed on a screen, a virtual image can not. Rays from both types can enter your eye, be refracted by your eye's lens and form a real image on your retina as @CarlWitthoft points out.
So it is not the case that a real image must be viewed on a screen. It can be viewed on a screen.
Any discussion of concave/convex mirrors needs to begin with a statement of the particular version of the mirror equation to be used, along with the convention for setting and interpreting the signs of focal lengths, and object/image positions.
For example, from http://scienceworld.wolfram.com/physics/MirrorFormula.html:
Unmentioned in this is the convention that virtual images and objects are found behind the mirror and have negative values of $d$.
In the particular example you present, the image formed by the primary mirror is a real image. If you put infinity for the object distance and a positive focal length, you find a positive image distance.
But when you insert a convex mirror, with a negative focal length, into the optical path, you must also consider the position of the real image (now an object) relative to the convex mirror. The object is behind the convex mirror; it is a virtual object, and its distance from the convex mirror is negative.
With appropriate positioning of the convex mirror, the formula will produce a positive value for the image distance. There will be real image formed in front of the convex mirror.
You've just designed a Cassegrain telescope...
Best Answer
to see the real inverted image "behind" the mirror is kind of an optical illusion. The picture is still in front of the mirror, but your eyes can not find a frame of reference, so it "sees" the picture at the usual place you see it in plane mirror, a virtual picture would not be inverted. to help your ey to see the picture, at the place where it really is, place som kind of mauve frame instead of the screen, sometimes it even helps if you put just your finger beside the place the screen was before and you see the real picture in the air beside your finger.