Why image formed in concave mirror appears wrong when light rays come from focus and center of curvature or parallel to it? [duplicate]

I found a problem were my calculations doesn't match with geometric drawing. I have an object placed at a position half of the focal legth of a concave mirror. I had drawn an image similar to one given below (Rays from center of curveture and parallel to principal axis) (like said in this question) but the image was not what it was supposed to be (They intersected much nearer to the mirror than they should). I tried with rays from focal point (reflacetd parallel to center) and parallel to principal axis (reflected through focus) but no avail. But when I drawn this (found in wikipedia), it was correct. Why my first two diagram didn't support the formula (1/f = 1/u + 1/v) ? Is drawing ray diagram from focal point (reflacetd parallel to center) and parallel to principal axis (reflected through focus) and the other method is wrong ?

What both your ray diagrams show is that a concave mirror can suffer from significant amounts of spherical aberration.

The mirror formula you use is derived on the assumption that the angles of incidence are very small and in terms of drawing a ray diagram one must therefore draw a concave mirror as being planar as shown below. Duplicate question found - Inconsistency in ray diagram and another Why do light rays intersect (or appear to intersect) at a specific point on reflection from spherical mirror?.

• Thank you for your recent edit. Sep 16 '21 at 22:09

Using the mirror formula

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$$

Here $$u = \frac{f}{2}$$

Substituting in mirror formula

$$\frac{1}{f} = \frac{2}{f} + \frac{1}{v}$$

Solving for $$v$$, we get

$$v = -f$$

i.e. the image is formed behind the mirror at the same distance as the focal length of the mirror. If you get any other result, it's wrong.

However if the aperture of the mirror is large, the rays do not converge at the focus of the mirror and hence, the mirror formula isn't valid in that case. Spherical mirrors are made with small apertures compared to the radii of curvature, so that the rays of light converge at the focus of the mirror. A large aperture will lead to spherical abberration.

• That's not the thing, the question is, why first two diagrams don't abide by this? Sep 11 '21 at 6:56
• Your object is probably much larger than the dimensions where the mirror formula is applicable. Large dimensions can cause abberrations where mirror formula isn't exactly valid. Sep 11 '21 at 6:58
• I get it, can you give the extended formula ? or a question related to such large object ? Sep 11 '21 at 6:59
• Unfortunately, I'm not an expert in optics. Maybe you should wait for another answer. Sep 11 '21 at 7:02