Referring to the electric room heater considered in problem \(5,\) calculate the intensity of IR radiation in \(\mathrm{W} / \mathrm{m}^{2}\) projected by the concave mirror on a person \(3.00 \mathrm{m}\) away. Assume that the heating element radiates \(1500 \mathrm{W}\) and has an area of \(100 \mathrm{cm}^{2}\), and that half of the radiated power is reflected and focused by the mirror.

Infrared radiation (IR) is a form of electromagnetic energy that occupies the wavelength spectrum just beyond what is visible to the human eye. Emitting heat that is felt but not seen, IR is commonly associated with warmth, as many sources of heat emit this type of radiation naturally.
In the context of your exercise, an electric room heater is the source of IR radiation. It's important to note that not all of the radiation from such a heater is infrared; however, typically, a significant portion is. IR heaters work by emitting radiation that is absorbed by objects, surfaces, and people, directly warming them instead of simply heating the air around them.
In educational settings, understanding IR radiation begins with recognizing its place in the electromagnetic spectrum and continues with considering its practical implications, such as its role in heating. These properties are essential when discussing the power and intensity of radiation emitted by devices like the one in the problem.

The power of radiation that is reflected by a surface, such as a concave mirror in an electric heater, refers to the portion of the energy that is sent back into the environment as opposed to being absorbed by the mirror. This reflected power is a critical concept because it dictates how effective a heating element is at redirecting energy towards a target, like a person in a room.
In the exercise, the heater radiates 1500 watts (W) of power, and the mirror reflects half of this power. To understand how much of that energy is effectively used to warm a person, we have to focus on the power subsequently projected. By capturing and refocusing the IR radiation, the concave mirror can concentrate the energy into a smaller area, resulting in increased heating efficiency. The amount of power that's actually reflected (750 W, in this case) is the starting point for calculating the intensity of the radiation experienced by someone nearby.

Radiation intensity, expressed in watts per square meter (W/m²), is a measure of the power of radiation per unit area. Calculating the intensity involves two primary steps: determining the power of the reflected radiation and the area over which this radiation is dispersed.
The solution you've worked through smartly separates these steps. First, it calculates the reflected power, which is the total energy available to be concentrated on a person. Then, considering that the concave mirror focuses this power onto a target, the exercise assumes this energy is distributed over a circular area with the person at the center. However, here it's important to clarify a potential misconception from the exercise: the area used in the intensity formula should refer to the cross-sectional area through which the radiation is passing, not the area of the heating element itself. Instead of using the heating element's area, we should be looking at the area over which the reflected power is spread when it reaches the person's position.
The intensity of the reflected radiation is then found by dividing the power of reflected radiation by the squared distance from the source, given that no other instructions about the shape of the beam were provided. This gives us an approximation of the intensity that a person would experience at a specific distance from the heater, illuminating the direct relationship between distance, power, and perceived warmth.