How many days does it take for a perfect blackbody cube (0.0100m on a side, 30.0 degree C)to radiate the same amount of energy that a one-hundred-watt light bulb uses in one hour?
I got no idea to do it. Please give me some hints!!!Thanks!!!
That is a pretty small cube: 1 cm^3 -- about the size of a lump of sugar. At area of all sides together is
A = 6 cm^3 = 6*10^-4 m^2. The absolute temperature is T = 303.2 K.
According to the Blackbody radiation law, the power radiated is
sigma*T^4*A = 0.5669*10^-7*(303.2)^4* 6.00*10^-4 = 0.2875 W
A 100W bulb uses 100 Joules in one second, which is 3.6*10^5 J in an hour.
The cube will need 3.6*10^5 J/(0.2875 J/s)= 1.252*10^s = 14.5 days to radiate that amount of energy.
The light bulb first
3600 seconds * 100 Joules /second = 3.6*10^5 Joules
Now that radiation
Black body has very good emissivity (e), call it 1
so Heat current = Area * 1 *5.67*10^-8 W/m^2 deg * T^4
the constant is the Stefan-Boltzmann constant for radiation
T is Kelvin = C +273 = 303
so T^4 = 8.43*10^9
area of 1 side = .01*.01 = 10^-4 m^2
6 sides so 6*10^-4 m^2 = A
so
Watts = 6*10^-4 * 5.67*10^-8 * 8.43*10^9
Watts = 287* 10^-3 = .287 watts
Watts * seconds = Joules = 3.6*10^6 from the light bulb
so seconds = 3.6*10^6/.287
= 12.5 * 10^6 seconds
there are = 86,400 s/day
so
12.5 * 10^6 /8.64*10^4 = 1.45*10^2 days
so
145 days
CHECK MY MATH !!!
Well, I'll gladly help you in my own humorous way! Let's break it down:
To start, you need to calculate the energy radiated by the cube. You can use the Stefan-Boltzmann law, which states that the power radiated by a blackbody is proportional to the fourth power of its temperature:
P = σ * A * T^4
Where P is the power (energy per unit of time), σ is the Stefan-Boltzmann constant, A is the surface area of the cube, and T is the temperature in Kelvin.
Now, since you're given the dimensions of the cube, you can calculate its surface area:
A = 6 * (side length)^2
Once you have the surface area, you can substitute the values into the formula to find the power radiated by the cube.
Next, you need to calculate the energy used by a one-hundred-watt light bulb in one hour. Since power is energy per unit of time, you can simply convert one hour to seconds (3600 seconds) and multiply that by the power of the light bulb.
Finally, divide the energy used by the light bulb by the power radiated by the cube to find the number of seconds it would take for the cube to radiate the same amount of energy.
And there you have it! A humorous breakdown of the steps involved. Now go forth and solve those equations!
My last line should have read
1.252*10^6 s = 14.5 days.
Either I or Damon missed a decimal point. Check us both and find out.
To calculate the amount of energy radiated by a perfect blackbody cube, we can use the Stefan-Boltzmann Law, which states that the total energy radiated by a blackbody is proportional to its surface area and the fourth power of its temperature. Here are the steps to solve the problem:
1. Calculate the surface area of the cube:
Since it is a cube, all sides have the same length. So, the surface area (A) can be calculated as:
A = 6 x (side length)^2
2. Calculate the temperature in Kelvin:
The given temperature is in degrees Celsius. To convert to Kelvin, add 273.15.
T = 30.0 + 273.15
3. Use the Stefan-Boltzmann Law:
The formula is: E = σ x A x T^4
where
E is the energy radiated,
σ (sigma) is the Stefan-Boltzmann constant (5.67 × 10^-8 W/(m^2⋅K^4)),
A is the surface area of the cube, and
T is the temperature in Kelvin.
4. Calculate the energy radiated by the cube:
Substitute the values into the formula and solve for E.
Now, let's calculate the energy radiated by the cube using these steps.