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==== Luminous Intensity ====


**Definition**\\
Luminous intensity describes the intensity of a light source in a specific direction, defined as the luminous flux emitted per unit solid angle. It is an intrinsic property of the light source and is independent of the source area and observation distance.\\
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(Image source: https://en.wikipedia.org/wiki/Radiant_flux#/media/File:Photometry_radiometry_units.svg)\\
Its mathematical expression is:
$$ I_v = \frac{\mathrm{d}\Phi_v}{\mathrm{d}\Omega} $$
where $I_v$ is the luminous intensity in the specified direction; $\Phi_v$ is the luminous flux emitted in the specified direction (unit: lm), and $\Omega$ is the solid angle containing that direction (unit: steradian, sr).

**Unit**\\
The SI unit of luminous intensity is the candela (symbol: cd), which is one of the seven base units in the International System of Units (SI). It is defined such that if a monochromatic light source emits radiation at a frequency of 540×10¹² Hz (corresponding to a wavelength of approximately 555 nm, the yellow-green light to which the human eye is most sensitive) and has a radiant intensity of 1/683 watts per steradian (W/sr) in a specified direction, the luminous intensity of the source in that direction is defined as 1 candela.\\

Combining the relationship between luminous flux and luminous intensity, its mathematical expression can be derived as:\\
$$\mathrm{cd} = \mathrm{lm} \cdot \mathrm{sr}^{-1}$$
That is, if the luminous flux of a light source within a solid angle of 1 sr is 1 lm, its luminous intensity in that direction is 1 cd.\\

**Core Derivations of Luminous Intensity**\\
**How to calculate luminous flux from the luminous intensity distribution?**\\
The distribution of luminous intensity with respect to the emission direction (\( \theta, \varphi \)) can be used to determine the luminous flux \( \Phi_v \) of a light source within a specific solid angle \( \Omega \):
\[ \Phi_v = \iint_\Omega I_v(\theta, \varphi) \sin\theta \, \mathrm{d}\varphi \, \mathrm{d}\theta \]

**How to obtain the visually perceived luminous intensity from physical radiation?**\\
Luminous intensity can be derived from the spectral radiant intensity distribution, with the formula:
\[ I_v = K_m \int_0^\infty I_{e,\lambda}(\lambda) V(\lambda) \mathrm{d}\lambda \]
where \( K_m \) is the maximum luminous efficacy, \( I_{e,\lambda}(\lambda) \) is the spectral radiant intensity at wavelength \(\lambda\), and \( V(\lambda) \) is the spectral luminous efficiency.

**Practical Scenario:**\\
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**Question:** Why is the luminous intensity of a desk lamp in the direction of the book higher than in other directions (such as towards the ceiling)?\\
Because the lampshade design of the desk lamp concentrates the light towards the book, making the luminous flux per unit solid angle in that direction much higher than in other directions.\\
Direction towards the book: High luminous intensity → more luminous flux per unit solid angle → more light received, so it appears brighter.\\
Other directions deviating from the book: Low luminous intensity → less luminous flux per unit solid angle → dimmer light.\\

**Frequently Asked Questions:**\\
**Does the luminous intensity of a light source change with distance?**\\
No, luminous intensity is an intrinsic property of the light source, determined by the light-emitting characteristics and light distribution design of the source itself, and is independent of the observation distance. What attenuates with distance is illuminance (the luminous flux received per unit area of the illuminated surface), which follows the inverse-square law: if the distance is doubled, the illuminance attenuates to 1/4 of its original value. In an ideal non-scattering medium, luminance also does not change with distance, as it describes the light-emitting characteristics of the light source's surface.\\

**What is the difference between luminous intensity and luminance?**\\
Luminous intensity is a physical quantity for point light sources. It describes the luminous flux emitted by the light source in a specific direction per unit solid angle, which is only related to the angular distribution of light and independent of the size of the light source. Luminance characterizes the visual brightness of an extended light source, which is jointly determined by the luminous intensity and the apparent size of the light source. Its value is independent of the observation distance and is only affected by the observation direction.\\