A 6.05-kg black cat and her four black kittens, each with mass o.792 kg, sleep s

Question

A 6.05-kg black cat and her four black kittens, each with mass o.792 kg, sleep snuggled together on a mat on a cool night, with their bodies forming a hemisphere. Assume the hemisphere has a surface temperature of 27.9°c, an emissivity of 0.970, and a uniform density of 1046 kg/m (a) Find the radius of the hemisphere (b) Find the area of its curved surface. m2 (c) Find the radiated power emitted by the cats at their curved surface (d) Find the intensity of radiation at this surface. You may think of the emitted electromagnetic wave as having a single predominant frequency. W/m2 (e) Find the amplitude of the electric field in the electromagnetic wave just outside the surface of the cozy pile N/C (f) Find the amplitude of the magnetic field. ?? (g) The next night, the kittens all sleep alone, curling up into separate hemispheres like their mother. Find the total radiated power of the family. (For simplicity, ignore the cats' absorption of radiation from the environment.)

Explanation / Answer

Part A

We know that

Mass = Volume*density

Suppose radius of hemisphere is r, then

Volume of hemisphere = 2*pi*r^3/3

density = 1046 kg/m^3

mass = 6.05 + 4*0.792 = 9.218 kg

Now

m = d*2*pi*r^3/3

r = (3m/(2*pi*d))^(1/3)

r = (3*9.218/(2*pi*1046))^(1/3)

r = 0.161 m

Part B

(Since Surface area of Sphere = 4*pi*r^2), So

Area of curved surface = 2*pi*r^2

Area = 2*pi*0.161^2 = 0.163 m^2

Part C

Power emitted is given by:

Power = Intensity*Area

Intensity will be

I = e*sigma*T^4

e = emissivity = 0.970

sigma = 5.67*10^-8 = Stefan-Boltzmann's constant

T = 27.9 C = 300.9 K

So,

P = I*A = e*sigma*T^2*A

P = 0.970*5.67*10^-8*300.9^4*0.163

P = 73.49 W

Part D.

Intensity = Power/Area

I = 73.49/0.163

I = 450.86 W/m^2

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