Hadi said:
Hello-
I am starting a project to get direct sunlight onto the house by placing convex mirrors on the outside stone fence of the backyard since it is the only place of my property that is not shaded by other houses in the neighborhood. For that, I need to calculate the size of the mirrors, given the distance from the stone fence on which they will be placed (on a swivel) to the house as well as the size of the glass doors that open up from the living room to the garden (I would rather that I cover the entire area of the doors with sunlight!). I tried to do some research on my own but I think I'm way over my head on this one. Is there some kind of formula I can use to calculate the area of the light that is reflected onto the house given the size of the mirror and the distance from the mirror to the house? Is there any other parameters that I have missed? Please feel free to ask me for more information if I have missed providing any!
Your project to use convex mirrors to direct sunlight into your house is quite innovative, but it does involve some complex optics. The behavior of light when it reflects off convex mirrors can be a bit tricky to calculate because convex mirrors diverge light rays, spreading them out over a larger area. This divergence means that the area of light that reflects onto your house from a convex mirror will be larger than the mirror itself, but also less intense.
To estimate the size of the mirrors needed, we need to understand a few key principles:
Mirror Equation: For convex mirrors, the mirror equation relates the object distance (
u
u), the image distance (
v
v), and the focal length (
f
f). The equation is given by
1
f
=
1
v
+
1
u
f
1
=
v
1
+
u
1
. However, for your purpose, this equation helps more with understanding the formation of images rather than the dispersion of light.
Field of View: The size of the area illuminated by the mirror will depend on its field of view, which is influenced by the mirror's curvature. Convex mirrors have a wider field of view than flat mirrors, allowing them to illuminate a larger area.
Geometry of Light Reflection: The geometry of how light reflects off the mirror and onto a target area (like your glass doors) is crucial. The angle of incidence equals the angle of reflection, but due to the convex nature of the mirror, rays diverge.
Distance and Size Relationship: The further the light has to travel from the mirror to the target, the more spread out (and thus, less intense) it will become. The size of the illuminated area on your house will depend on the mirror's curvature and the distance from the mirror to the house.
To estimate the size of the mirrors, consider these steps:
Mirror's Curvature (Focal Length): The curvature of the mirror determines its focal length, which affects how much it will spread out the light. A mirror with a shorter focal length will spread light more than one with a longer focal length.
Distance to the House: The further the mirror is from the house, the larger the mirror needs to be to illuminate the same area, due to the divergence of light rays.
Desired Illuminated Area: You want to cover the area of the glass doors. Knowing the dimensions of these doors helps in determining how wide the beam of light needs to be when it reaches the house.
A simple way to start is by using a practical approach, considering the angle at which sunlight hits the mirror and the angle needed to reflect it towards your doors. However, without specific measurements (size of the doors, distance from the mirror to the doors, and the amount of sunlight you wish to redirect), it's challenging to provide a precise formula.
As a practical experiment, you could use a small, movable mirror to test how different sizes and angles affect the sunlight's coverage on your doors. This hands-on method could give you a rough idea of the mirror size needed before committing to larger, more expensive mirrors.