Question

Consider a room heater that uses a concave mirror to reflect infrared radiation from hot coils. The mirror has a radius of curvature of 47.0 cm and produces an image of the coils 2.60 m away from the mirror. Calculate the intensity of IR radiation in W/m2 projected by the concave mirror on a person 2.60 m away. Assume that the heating element radiates 1,250 W and has an area of 130 cm2, and that half of the radiated power is reflected and focused by the mirror.

Answer 1

From the information given,

radius of curvature, r = 47 cm = 47/100 = 0.47 m

focal length, f = r/2 = 0.47/2 = 0.235

image distance, di = 2.6

We would find the object distance by applying the formula,

1/di + 1/do = 1/f

1/2.6 + 1/do = 1/0.235

1/do = 1/0.235 - 1/2.6

1/do = 3.871

do = 1/3.871 = 0.258 m

Magnification = di/do = 2.6/0.258 = 10

Power reflected by the mirror = 1250/2 = 625

Given that magnification of object is 10 times the magnification of image,

Area magnification = 10 x 10 = 100

Area of heating element = 130 cm^2

1 cm^2 = 0.0001 m^2

130 cm^2 = 130 x 0.0001 = 0.013 m^2

Area of object x area magnification = 100 x 0.013 = 1.3 m^2

Area of image = 1.3m^2

Intensity of reflected radiation is

I = P/A

I = 625/1.3

I = 480.77 W/m^2

The intensity is 480.77 W/m^2