Answer:
a
[tex]\frac{P_1}{P_2} = 1.188 \ m [/tex]
b
[tex] Z = 1.08995 [/tex]
Explanation:
From the question we are told that
The filament temperature of the first bulb is [tex]T_1 = 2100 ^o C = 2100 + 273 = 2373 \ K [/tex]
The filament temperature of the second bulb is [tex]T_3 = 2 0008^o C = 2000 + 273 = 2273 \ K[/tex]
Generally according to Stefan-Boltzmann law the power emitted by first bulb is mathematically represented as
[tex]\frac{P}{A} = \sigma T^4_1 [/tex]
Here [tex]\sigma[/tex] is the Stefan-Boltzmann with value [tex]\sigma = 5.67*10^{-8} \ W . m ^{-2}. K ^{-4}[/tex]
So
[tex]\frac{P_1}{A_1} = (5.67*10^{-8}) (2373)^4 [/tex]
=> [tex]P_1 = 1797935 A_1 [/tex]
Generally according to Stefan-Boltzmann law the power emitted by second bulb is mathematically represented as
[tex]\frac{P_2}{A_2} = \sigma T^4_2 [/tex]
[tex]\frac{P_2}{A_2} = (5.67*10^{-8}) * (2273)^4 [/tex]
=> [tex]P_1 = 1513494 A_2 [/tex]
Given that the two bulbs are identical we have that
[tex]A_1 = A_2 = A[/tex]
So
The ration is mathematically represented as
[tex]\frac{P_1}{P_2} = \frac{1797935* A}{1513494 *A}[/tex]
=> [tex]\frac{P_1}{P_2} = 1.188 \ m [/tex]
Generally the area is mathematically represented as
[tex]A = \pi r^2[/tex]
Recall that
[tex]A_1 = A_2 = A[/tex]
=> [tex]\pi r^2_1 = \pi r^2_2 = \pi r^2[/tex]
=> [tex]r^2_1 = r^2_2 = r^2[/tex]
Now if the radius of the cooler bulb is increase by a factor Z (i.e Z * r )then the area of the cooler bulb [tex]A_2 [\tex] becomes
[tex] A_2 = (Zr)^2 [/tex]
=> [tex] A_2 = Z^2*r^2 [/tex]
Here Z is the factor by which it is made thicker
So For [tex]\frac{P_1}{P_2} = \frac{1797935* A}{1513494 *A}[/tex]
[tex] 1.188 = \frac{1797935* r^2}{1513494 *Z^2*r^2}[/tex]
=> [tex] Z = 1.08995 [/tex]
