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I've noticed an odd phenomenon with an adjustable-focus LED flashlight that I've bought.

If I shine it at the ceiling of my room at the widest possible angle, the walls get noticeably brighter than if I shine it at the ceiling of my room at the narrowest possible angle.
Of course, in both cases, the total light emitted is the same, and so is the center of the lit area.

Shouldn't the walls be lit by the same amount in both case?

Why does the surface area of reflection affect the outcome when the amount of light is the same?

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Think of the rays at the extremities of the cone of light. The angle of incidence on the surface of the ceiling from these rays and so the scattering angles from the surface of the ceiling will be greater that those rays which hit the ceiling at smaller angles (you concentrated cone).
So there will be more light heading towards the walls with your wide angle beam.
As a test of my suggestion look down and see what happens to the floor?

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What you are observing is the apparent total brightness reported to you via the optical sensing portion of your brain. A lot of things go into that subjective feeling: how wide your iris opens, how many rods/cones in the retina are stimulated, and what the relative peak and minimum intensities are in your field of view. If you in fact used a calibrated light meter, you would find, with some difficulty (need to measure at all angles looking at all wall points) that the total reflected radiance is the same.

So by and large what you're reacting to is the total amount of wall area that is illuminated, rather than the actual level of ilumination at any given point.

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