The energy radiated per solid angle is often referred to as radiant intensity (I) and is a measure of how much energy is emitted by a light source in a particular direction within a given solid angle. Radiant intensity is expressed in watts per steradian (W/sr).

Mathematically, the radiant intensity (I) is defined as the radiant flux (Φ) per unit solid angle (Ω)

I = dΦ/dΩ

Where:

I is the radiant intensity (in watts per steradian, W/sr).
ΦdΦ is the differential radiant flux (in watts, W) emitted within a differential solid angle ΩdΩ.

To put it simply, radiant intensity tells you how much energy is radiated per unit solid angle in a specific direction. It is commonly used in radiometry and photometry to characterize the brightness or power output of light sources.

For example, a laser beam may have a high radiant intensity because it emits a concentrated and intense beam of light within a small solid angle. On the other hand, a diffuse light source, like a light bulb, may have a lower radiant intensity because its light is spread out in many directions, resulting in a lower radiant intensity for any individual direction.

The radiant intensity is an important parameter when considering the behaviour of light sources, the design of optical systems, and various applications in optics and illumination engineering.

Frequency by a charged particle moving instantaneously in circular path.

Consider a charged particle moving in curved path. A small path length of path can be considered as circle. Let V be speed and centre of circular arc be on y axis.

Let v be the speed and centre of circular arc be on yaxis. Let 𝜌 is the radius of arc then the angle described about the centre is vt/𝜌 in time 't'. The energy radiated per solid angle per frequency range is

Let Ell and E⟂ be unit vectors along y axis and tangent to the path then v has two components

hence equation 2 gives

Consider the argument of exponential in 1,

Substituting 3 and 4 in 1

The Airy integrals are defined as

The energy radiated per frequency per angle

𝜃 = 0, 𝜉 >> 0

If you have the angular frequency of a charged particle moving in a circular path, you can use the formula above to calculate the frequency f of the radiation emitted by that particle as a result of its circular motion.

This note is a part of the Physics Repository.

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