intensity of a light equation point source

However, the distance r which the law relates to has to be measured from the position of the virtual point source.īy definition, a Lambertian surface either emits or reflects radiation with constant radiance (L e) in all directions of a hemisphere (s. Fig. 2). Remark: Since a virtual point source located at the cone’s vertex produces the same spatial radiation distribution as the flashlight’s bulb together with its concave mirror, the “inverse square law” applies for this configuration. Thus, the cone defines a solid angle given by Ω =Īnd the flashlight’s radiant intensity amounts to I = Where r is the distance of the circle from the cone’s vertex. Following the definition of solid angle and approximating the area of the spherical calotte using the area of a circle with a 5 cm radius (= 0.05 m), we get Ω = If we assume that irradiance is constant all over this circle and we neglect the fact that the surface is not perfectly even hence not strictly perpendicular to the beam, we can calculate the irradiance at a 25 cm distance from the flashlight’s front window: E = radiant power impinging upon a surface or area of this surface =įig. 1: Calculating the irradiance caused by a flashlightī) In order to determine the flashlight’s radiant intensity, we have to determine the solid angle determined by the cone.

Note that the flashlight does not emit light symmetrically in all directions and the equations derived in Example 1 can therefore not be used.Ī) At a distance of 25 cm from the flashlight’s front window, the whole radiant power of 200 mW (= 0.2 W) impinges on a circle with a 0.05 m radius. Assuming that the mirror reflects without any losses a uniform distribution of power over the cone is generated. In a simple flashlight, a concave mirror reflects light from a small bulb (radiant power Φ = 200 mW) into a divergent cone (see Fig. 1). Fluorescent tubes are a good example in this case. However, when the source cannot be equated with a point source and every point of the source emits light in more than a single direction, the “inverse square law” can no longer be applied. In other cases, a source with considerable geometric dimensions might possibly be replaced by a “virtual” point source, for which the “inverse square law” would still apply at a distance r from this virtual point source (see Example 2). However, it only holds true for distances much larger than the geometric dimensions of the source, which allows the assumption of a point source. Remark: The proportionality of E to r² is generally described with the “inverse square law”. Which is identical with the result above. Radiant power impinging upon a surface or area of this surface Thus, irradiance E of a surface at a certain distance r and oriented perpendicular to the beam can be calculated from its definition: E e = This result can also be obtained by the following argument:Īt distance r, all the radiant power Φ e,source emitted by the source passes through the surface of a sphere with radius r, which is given by 4r²π. Because the light source emits light symmetrically in all directions, the irradiance has the same value at every point of this sphere. The irradiance at distance r therefore amounts to E e = As a rule of thumb, this approximation is justified if distance r is at least 10 times larger than the dimensions of the light source.Ī) Since the source emits light symmetrical in all directions, its radiant intensity is equal for all directions and amounts to I e =ī) An infinitesimal surface element dA at a distance r and perpendicular to the beam occupies the solid angle dΩ =Īnd thus the infinitesimal radiant power d Φ e,imp impinging onto dA can be calculated by d Φ e,imp = I dΩ =

If we are interested in the characteristics of this source at a distance (r) that is much larger than the geometric dimensions of the source itself, we can neglect the actual size of the source and assume that the light is emitted from a point. Its radiant power equals Φ e,source = 10 W. A small source emits light equally in all directions (spherical symmetry).

Intensity of a light equation point source