Photometry

1. What are the important concepts of photometry?

Solution:

Photometry is a part of optics that deals with the energy transmitted by light and the visual effects of light.

The most important concepts of photometry are:
a) solid angle
b) luminous flux
c) luminous intensity of a source
d) illumination

Solid angle (Ω) is a cone with its vertex at a point light source, which on a sphere with radius r cuts out an area S.

\Omega = \frac{S}{r^2}, \quad [\Omega] = \text{steradian} = sr

Luminous flux (Φ) is the radiant energy passing through an area S in time t.

\Phi = \frac{E_s}{S \cdot t}, \quad [\Phi] = W \cdot m^{-2} = lumen = lm

Luminous intensity (I) is the luminous flux Φ emitted into the solid angle Ω.

I = \frac{\Phi}{\Omega}, \quad [I] = \text{candela} = cd, \quad S = 4 \pi r^2

Illumination (E) is the luminous flux falling on a unit area.

E = \frac{\Phi}{S} = \frac{lm}{m^2} = lux = lx

E = E_0 \cos \alpha \quad \Rightarrow \quad \frac{E}{E_0} = \cos \al

where
$\alpha$ = the angle between the normal of the surface and the direction of light propagation.

2.Define:

candela
lumen
lux

Solution:

One candela (cd) is 1/600,000 of 1 m² of the surface of an absolutely black body at the freezing point of platinum (1773⁰C) at normal pressure (101,325 Pa). An absolutely black body is a fictitious body that would absorb all incident radiation.
One lumen (lm) is the luminous flux emitted by a point source of luminous intensity one candela into a solid angle of one steradian.
A surface has an illuminance of one lux (lx) if a luminous flux of one lumen falls on 1 m² of that surface.

3. The human eye perceives a light sensation at a minimum retinal illuminance of E=2·10^-9 lx. From what maximum distance, under ideal conditions (absolute darkness), can we observe a candle flame with luminous intensity I = 1 cd?

Solution:

Analysis:

The candle flame can be observed from a maximum distance of 22.4 km.

4.A photometer is illuminated on one side by a light bulb with I₁ = 50 cd at a distance r₁=0.6 m. On the other side, at a distance r₂ = 0.96 m, there is a bulb of unknown luminous intensity (I₂ = ?). What is its luminous intensity if the photometer is equally illuminated from both sides?

Solution:

Analysis:

The luminous intensity of the second bulb is I₂ = 128 cd.

5.Above a circular tabletop with radius r = 0.8 m hangs, at height h₁ = 2 m, a light bulb with luminous intensity I = 50 cd. Determine the illuminance

a) at the center of the tabletop
b) at the edge of the tabletop

Solution:

Analysis:

I = 50 cd, r = 0.8 m, h₁ = 2 m, h₂ = ? = distance from the bulb to the edge of the tabletop. The quantities r, h₁, h₂ form a right triangle with hypotenuse h₂. Angle α is the acute angle opposite the leg r.

The illuminance at the center of the tabletop is 12.5 lux, and at the edge it is 10 lux.

6.Two bulbs with luminous intensities I₁ = 80 cd and I₂ = 200 cd are placed at height h = 2.5 m above a horizontal tabletop. Their mutual distance is d = 3 m. What is the illuminance on the table at the point where the rays from the first bulb strike?

Solution:

Analysis:

The resulting illuminance from both bulbs is E = 22.8 lux.

7. The illuminance of a horizontal surface of the Earth when the Sun’s altitude above the horizon is 45⁰ is E₁ = 80,000 lx. Determine the illuminance of the Earth’s surface when the Sun’s altitude is 25⁰.

Solution:

Analysis:

The illuminance of the Earth’s surface at a Sun altitude of 25⁰ is 47,800 lux.

8. A wall is illuminated by two identical candles placed right next to each other at a distance of 0.8 m from the wall. By what distance x should one candle be moved closer to the wall if the other goes out (I₂ = 0) so that the wall is illuminated the same as before?

Solution:

Analysis:

The candle must be moved 23 cm closer to the wall.

9.A circular area with radius r = 3 m is illuminated by a reflector from a distance h = 60 m. The illuminance of the area is E = 4 lx. Calculate:

a) the luminous intensity of the source
b) the luminous flux incident on the area

Solution:

Analysis:

The luminous intensity of the reflector is I = 1.44 cd, the luminous flux Φ = 113 lm.

10. Two light sources with luminous intensities I₁ = 25 cd and I₂ = 49 cd are placed at a distance d = 1 m. At what point on the line joining them should a screen be placed so that it is equally illuminated by both sources? (I₂ is to the left of I₁)

Solution:

Analysis:

There are two possible positions for the screen:

0.42 m from I₁ and 0.48 m from I₂,
-2.5 m from I₁ (to the right of I₂) and 3.5 m from I₂.

Physics

Photometry

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