Mistakes about illuminance

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In summary: Thus, when calculating the illuminance, you should divide by only half the surface area of the sphere, not the entire surface area. This will give you the same result as the book. In summary, the question asks for the illuminance of the floor directly below a spherical lamp with a radius of 6.0 cm and a luminance of 2.0*10^4 cd/m2, located at a height of 3 metres. The book's solution is I=25.13 lux, while the student's solution is I=50.27 lux. However, the student's solution is incorrect because they are using the total luminance of the sphere, while the book's solution is using only half of the sphere's
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Hello everybody,
this question is about an exercise but I post it here because it is not homework, it is an exercise that I've done to learn by myself. I hope it is ok.
It is an exercise from Irodov book (exercise 5.12)
The exercise says:
A small spherical lamp, uniformly luminous with radius R= 6.0 cm is suspended at an height h of 3 metres above the floor;
The luminance of the lamp is L=2.0*10^4 cd/m2, indipendent of direction.
Find the illuminance of the floor directly below the lamp.

the solution that the book (and my teacher) gives is

[tex]I= \pi\frac{R^{2}}{h^{2}} L = 25.13\, \, lux [/tex]


I tried to solve it in this way:
the symmetry of the problem gives us many advantages; we can obtain the total luminous flux of the lamp by multiplying the luminance by the total surface and by 2 pi steradians (half of the maximum solid angle, because i assume the sphere doesn't radiate inside itself)

[tex]F=L (2 \pi) (4 \pi R^{2} )= L (8 \pi^{^2} R^{2} )[/tex]

then, to have the illuminance of the floor just below the lamp, we can divide the total flux by the area of the sphere having radius h:

[tex]I= L (8 \pi^{^2} R^{2} )/ (4 \pi h^{2})= 2 \pi L \frac{R^{2}}{h^2}=50.27\; lux[/tex]

but as you can see it is exactly twice the solution given by the book.
I suppose i am wrong, but i cannot understand why. can you help please?
Thank you in advance, sorry for my english
 
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.Your formula is correct, but the reason why you get twice the results of the book is because the luminance you are given is already the one of the hemisphere facing downwards. The total luminance of the sphere should be twice that value.
 

What is illuminance and why is it important?

Illuminance is the measure of the amount of light that falls on a surface. It is important because it affects how we perceive our surroundings and can impact our visual comfort and productivity.

What are some common mistakes people make when measuring illuminance?

One common mistake is not taking into account the distance between the light source and the surface being measured. This can greatly affect the illuminance levels. Another mistake is not considering the directionality of the light source, as light intensity can vary based on the angle of incidence.

How do you convert between different units of illuminance?

The most commonly used unit of illuminance is lux, but other units such as footcandles and lumens per square meter may also be used. To convert between these units, you can use conversion factors or online calculators.

What factors can affect illuminance levels?

The amount of light that reaches a surface is affected by various factors, including the distance and direction of the light source, the reflectance of the surface, and any obstructions or shadows that may be present.

How can mistakes in measuring illuminance be avoided?

To avoid mistakes, it is important to use the correct measuring equipment and techniques. It is also helpful to have a good understanding of the factors that can affect illuminance levels. Regularly calibrating equipment and taking multiple measurements can also help to ensure accurate results.

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