How is the angular magnification of a magnifying glass related with how much bigger the image appears to the user? Is it simply the same? If yes, why? (Let's assume that the object is placed in the focal plane for simplicity)
I was just playing around a bit with a magnifying glass with a focal length of $12\,$cm (so the angular magnification is about $2$). When I measured the size of the image on the surface of the lens, it was about $2.5$ times the size of the original object.
Best Answer
It is not clear to me what you meant by " I measured the size of the image on the surface of the lens".
You calculated value of approximately 2 was done on the assumption that the object was in the focal plane of the convex lens of focal length $f = 12$ cm.
If the height of the size is $h$ the best the naked eye can do is to view the object at the least distance of distinct vision $D$ which is generally taken to be $25$ cm. The angular magnification is $\frac {25}{12} \approx 2$.
The highest magnification is obtained when the final virtual image is at the least distance of distinct vision which would require the object to be approximately $8$ cm from the lens. This would give an angular magnification of $\frac {25}{8} \approx 3$.
To measure the magnification you need to set your lens $12$ cm from a grid and observe the image of the grid through the lens which is formed at infinity .
At the same time you need to observe another grid which is 25 cm from the (other?) eye.
This is very difficult to do.
Much easier is setting up the lens so that the final image is at the near point and I have done a simple experiment which produced a surprisingly good result.
I found a hand magnifier whose focal length was approximately $5$ cm and set it up to be about $4$ cm from the lens so that the virtual image would be about 25 cm from the lens.
I then put another grid 25 cm from the lens as shown in the photograph.
What was pleasing was that the iPhone simultaneously brought into focus the grid viewed through the lens and the grid 25 cm below the lens.
Note that the grid 4 cm from the lens was out of focus and "bigger" than the grid 25 cm from the lens.
If I had used that as my direct view grid as the reference the magnification found would have been in error.
10 magnified small squares were equal to 63 unmagnified small squares which gave a magnification of approximately $6$ which is not bad when compared with the theoretical value of $\frac {25}{4} \approx 6$.
So perhaps it is worth having another go at measuring the magnification of your $12$ cm lens noting that it is not only the focal length but the optical configuration which determines the magnification?
Later
The magnification $M$ of a magnifying glass is defined as
$$M = \dfrac{\text{angle subtended by image of object when 25 cm from the lens}}{\text{angle subtended by object when 25 cm from the naked eye}} = \dfrac {\alpha '}{\alpha}$$
The HyperPhysics article Simple Magnifier gives some more theory.