Quick and easy way to measure magnication of a concave mirror

[Lighting_man]

Hi,

We're offered shaving mirrors by suppliers and sometimes their claims to the magnification are doubtful.

I don't wish to upset any customers by giving incorrect information.

So is there a practical way in which I can test the magnification myself? i.e. measuring the distance of the bulge inward compared with the diameter of the glass?

Please be aware that I do not have a whole host of equipment (no flux capacitor etc) but have some accurate rules and a digital caliper.

 

[minger]

Doing a quick google turns up

http://www.glenbrook.k12.il.us/gbssci/phys/CLass/refln/u13l3e.html
http://www.glenbrook.k12.il.us/gbssci/phys/class/refln/u13l3f.html

It looks like as a test, you can place an object at distance X. Move the mirror away until no image appears. This is your focal length.

With the focal length and the actual objects distance away, you can calculate the image distance by:
$$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$
Where o indicates object, i is image, and f is the focal length. Magnification is then:
$$M = -\frac{d_i}{d_o}$$

OK, I think I've found another way. From the Wikipedia page for focal length http://en.wikipedia.org/wiki/Focal_length, we have the focal length as a function of the radius of curvature:
$$f = -\frac{R}{2}$$
So, assuming you have a spherical mirror, I 'think' here's what can be done geometrically.

Measure the arc length of the mirror (l), along with the diameter (D) (as you are calling it) and the depth (t) of the mirror. The "angle" of the mirror can be found by
$$\theta = \arctan\frac{t}{D/2}$$

Nope...better yet, some more Google (God bless it) shows that this "depth" of mirror is referred to as the sagitta. It is related to the "diameter" and radius by:
$$t = r - \sqrt{r^2 - (D/2)^2}$$

Therefore, the radius of the curve is related as:
$$r = \frac{s^2 + (D/2)^2}{2s}$$

From there, divide by 2 to get focal length, then relate by approximate distance of object as mentioned to get magnification...whew.

 

[Lighting_man]

Oh wow!

Okay so if the diameter of the mirror is 200mm (a 100mm radius)

and the sagitta is 10mm and the glass is parabolic.

Oh I'm pretty poor at brackets

Any chance of a more layman version?

 

[Bob S]

You could use the same measurement system as is used by eyeglass manufacturers. A one-diopter lens will focus the Sun to a point in one meter. A four-diopter lens will focus the sun to a point in 1/4 meter, or about 10 inches. I suspect your shaving mirror will focus the Sun to a point in about 10 inches, hence four diopters.
Bob S

 

[minger]

If your mirror is parabolic, the equation is even easier.
$$r = \frac{(D/2)^2}{2s}$$
So, plugging in gives you a radius of curvature:
$$r = \frac{100mm}{2*10mm} = 500mm$$

The focal length is then 250mm. If we assume that the mirror is held 200mm away from the object, then:
$$\frac{1}{250mm} = \frac{1}{200mm} + \frac{1}{d_i}$$
or:
$$d_i = 1000mm \rightarrow M = 5.0$$

...I think...