Reflection from a small spherical mirror

[bob012345]

I was testing a small spherical mirror with sunlight and wondered about something. The size of the mirror is 2.5cm. The spot size of the reflected light grows over larger distances but it doesn't seem linear. For example, at a meter or less, the spot is very close to the mirror size but at a few times that it's about twice the size. I was expecting the image to be essentially the size of the mirror with virtually parallel light from the sun which I have seen with larger mirrors. I tried the opposite experiment where I attached the mirror backwards to a large planar mirror and looked at the size of the dark spot. That got smaller to about half size at larger distances. I wouldn't think it's a true wave effect such as diffraction. Is it? I wouldn't think it's due to the finite angular image of the sun. Is it? Another clue, a larger circular mirror of about 10cm diameter shows only a small enlargement of about 1cm around the spot size over larger distances suggesting my hunch it's the solar image not being a point. If it's due to either of these I would think it's the angular size of the sun since my primitive calculations seem to suggest a reasonable close correlation. But that should, be linear. Maybe it is diffraction from the edge. Any thoughts? Thanks.

 

[sophiecentaur]

There is a simple formula for a concave mirror here. The calculations are not linear as they involve the inverse of distances.

 

[Drakkith]

I was expecting the image to be essentially the size of the mirror with virtually parallel light from the sun which I have seen with larger mirrors.

The size of an in focus image depends mostly upon the focal length of the mirror, not the size.

I tried the opposite experiment where I attached the mirror backwards to a large planar mirror and looked at the size of the dark spot. That got smaller to about half size at larger distances. I wouldn't think it's a true wave effect such as diffraction. Is it?

If I understand your setup, no. The size of the shadow of the mirror can be described by geometrical optics (ray optics), where we don't even consider diffraction.

I wouldn't think it's due to the finite angular image of the sun. Is it?

The Sun is an extended light source, not a point source. This causes the edges of the shadow to taper off instead of being an abrupt border. The actual size of the shadow depends on the size of the object and the distance between it and the surface. The fact that the Sun is an extended source also means that as you move the object away from the surface, eventually you'll reach a point where the angular size of the object, as viewed from the surface, is smaller than that of the Sun. The object can no longer block out all of the light from the Sun and any particular point on the surface will see at least part of the Sun. As you continue to move the object away from the surface the shadow decreases in intensity as more and more of the Sun is uncovered.

Another clue, a larger circular mirror of about 10cm diameter shows only a small enlargement of about 1cm around the spot size over larger distances suggesting my hunch it's the solar image not being a point.

A spherically curved mirror will only focus light to a sharp image when the distance between the surface and the mirror is equal to the focal length (for collimated light, which Sunlight is a very good approximation of). If you move the mirror further from the surface, the image becomes blurred. This blur increases as you move the mirror away.

 

[bob012345]

The size of an in focus image depends mostly upon the focal length of the mirror, not the size.
If I understand your setup, no. The size of the shadow of the mirror can be described by geometrical optics (ray optics), where we don't even consider diffraction.
The Sun is an extended light source, not a point source. This causes the edges of the shadow to taper off instead of being an abrupt border. The actual size of the shadow depends on the size of the object and the distance between it and the surface. The fact that the Sun is an extended source also means that as you move the object away from the surface, eventually you'll reach a point where the angular size of the object, as viewed from the surface, is smaller than that of the Sun. The object can no longer block out all of the light from the Sun and any particular point on the surface will see at least part of the Sun. As you continue to move the object away from the surface the shadow decreases in intensity as more and more of the Sun is uncovered.
A spherically curved mirror will only focus light to a sharp image when the distance between the surface and the mirror is equal to the focal length (for collimated light, which Sunlight is a very good approximation of). If you move the mirror further from the surface, the image becomes blurred. This blur increases as you move the mirror away.

Thanks. I wasn't clear in my introduction. The mirror is a spherical shape, round but not a spherical mirror in the sense of having a focal point. I should have been more clear on that, very sorry. It's just a round shaped plain mirror so I don't believe it has a focal length. I guess we can forget the round shape. I wonder if it's the same reason shadows are blurry. I agree that diffraction wouldn't be at play.

 

[bob012345]

There is a simple formula for a concave mirror here. The calculations are not linear as they involve the inverse of distances.

Thanks. Very sorry I wasn't clear. It's just a plain mirror that happened to have a round shape. As an aside, I was also playing with one if these concave mirrors and forming the image on a screen of a light bulb in this case. I saw multiple overlapping images due to multiple reflection since the mirror was a low quality cosmetic use mirror. So the mirror doesn't have a true uniform shape which caused distortion. It seemed to be quadrant related.

 

[jbriggs444]

Thanks. Very sorry I wasn't clear. It's just a plain mirror that happened to have a round shape.

That's "plane" mirror that happens to have a round shape.

If you were using a point source at infinity and reflecting it from a one inch diameter planar mirror onto a screen that was (let's say) six feet away, the expected illuminated region would be one inch in diameter. This is exactly the same thing as cutting a one inch hole in a black wall and letting the sun shine through onto a screen six feet behind the wall.

But the sun is not a point source. The illuminated spot will reflect the angular size of the sun. At six feet, one would expect the spot to bloom by an extra 1/3 of an inch radius to about 1 2/3 inch diameter. [Take the sine of the half angle and multiply by six feet to get the bloom on one side]

 

[bob012345]

That's "plane" mirror that happens to have a round shape.

If you were using a point source at infinity and reflecting it from a one inch diameter planar mirror onto a screen that was (let's say) six feet away, the expected illuminated region would be one inch in diameter. This is exactly the same thing as cutting a one inch hole in a black wall and letting the sun shine through onto a screen six feet behind the wall.

But the sun is not a point source. The illuminated spot will reflect the angular size of the sun. At six feet, one would expect the spot to bloom by an extra 1/3 of an inch radius to about 1 2/3 inch diameter. [Take the sine of the half angle and multiply by six feet to get the bloom on one side]

Thanks. So if I understand correctly, a 10 inch diameter mirror reflected spot would also 'bloom' by the same 2/3 inch overall to 10.6 inch radius not 16.6 inches and a 100 inch to 100.67 and so on. So it is the suns angular size at play. A perfectly collimated beam would make an exact size spot assuming no other effects.

 

[jbriggs444]

Thanks. So if I understand correctly, a 10 inch diameter mirror reflected spot would also 'bloom' by the same 2/3 inch overall to 10.6 inch radius not 16.6 inches and a 100 inch to 100.67 and so on. So it is the suns angular size at play. A perfectly collimated beam would make an exact size spot assuming no other effects.

Right.

Assuming that my calculations are right -- and they pass the sanity check of morning sun on floor reasonably well.

 

[sophiecentaur]

The effect of this circular plane mirror will be the same as what you get with a circular hole in a large screen with the sun shining perpendicularly onto it. The resulting shadow is perhaps easier to think of. You have an Umbra and a Penumbra. I looked briefly for a suitable image on Google but was flooded with irrelevant stuff about black holes etc. There is sure to be one somewhere though. A DIY diagram would be easy enough to sketch.

 

[bob012345]

The effect of this circular plane mirror will be the same as what you get with a circular hole in a large screen with the sun shining perpendicularly onto it. The resulting shadow is perhaps easier to think of. You have an Umbra and a Penumbra. I looked briefly for a suitable image on Google but was flooded with irrelevant stuff about black holes etc. There is sure to be one somewhere though. A DIY diagram would be easy enough to sketch.

Yes, thanks. That's a good idea.