Infrared telescopes are very sensitive to increases in temperature. If the
temperature rises above a certain point, the telescope is blinded because it
becomes a bright source of infrared light itself (if that doesn�ft immediately
make sense, it is similar to looking at a star with the naked eye during the
daytime). With this in mind, you need to design your circuit in such a way
that the load does not heat up too much. To do this, use the formula P =
ƒÐAT^4, where P is the power dissapated by the load, A is the surface area of
the load, T is the temperature in Kelvin, and ƒÐ = 5.67�~10^−8 Wm^2K^−4 is
the Stefan-Boltzmann constant. If we need to construct the circuit so that
the load never gets hotter than 120 Kelvin, what is the minimum value we
can use for the resistance of the load? Use the values L = 1 mH, C = 1 ƒÊF,
A = 10 cm2, and the maximum voltage across the source is 12 V.
You talk about a circuit but do not show it. You also don't say how the circuir realtes to the telescope.
Where is the capacitor and inductance for which you provide C and L values? They don't seem to matter at all. Apparently you are supposed to make sure a resistor stays below 120 K. Assuming the resistor is in vacuum, and only able to lose heat by radiation, then you can compute the I^2 R power dissipation and set it equal to the radiative power loss, for which they provide the power and emissivity.
All the talk about infrared telescopes is just there to confuse you, but it IS very important to keep the temperature of all objects SEEN by such telescopes low, including the mirror and the detector.
To find the minimum value for the resistance of the load, we can use the formula P = ƒÐAT^4, where P is the power dissipated by the load, A is the surface area of the load, T is the temperature in Kelvin, and ƒÐ = 5.67 × 10^-8 Wm^2K^-4 is the Stefan-Boltzmann constant.
We can start by rearranging the formula to solve for resistance. Since power is equal to voltage squared divided by resistance (P = V^2 / R), we can substitute P / A into the formula for power.
P = ƒÐAT^4
V^2 / R = ƒÐAT^4
R = V^2 / (ƒÐAT^4)
Now, let's plug in the given values:
- Maximum voltage across the source (V) = 12 V
- Surface area of the load (A) = 10 cm^2 = 0.01 m^2
- Maximum temperature (T) = 120 Kelvin
- Stefan-Boltzmann constant (ƒÐ) = 5.67 × 10^-8 Wm^2K^-4
R = (12^2) / (5.67 × 10^-8 × 0.01 × (120^4))
Calculating this expression, we get:
R ≈ 7.59 × 10^8 ohms
Therefore, the minimum value we can use for the resistance of the load is approximately 7.59 × 10^8 ohms.