A heat lamp produces 20.0 watts of power at a wavelength of 6.7 μm .
I don't see a question.
To understand how to answer this question, let's break it down step by step.
First, let's define the given information:
- Power: 20.0 watts
- Wavelength: 6.7 μm (micrometers)
Now, we need to understand the relationship between power and wavelength for a heat lamp. This relationship is determined by Planck's law and the Stefan-Boltzmann law.
Planck's law states that the power emitted by a black body radiator (like a heat lamp) at a specific wavelength is proportional to the intensity of the radiation at that wavelength. In mathematical terms, the power emitted (P) is equal to a constant (C) multiplied by the wavelength (λ) to the power of 5, divided by the exponential of a constant (D) multiplied by the wavelength and temperature (T) minus 1.
Stefan-Boltzmann law, on the other hand, describes the total power radiated by a black body radiator as being proportional to its temperature (T) raised to the fourth power. It states that the total power radiated (P_total) is equal to another constant (σ) multiplied by the temperature (T) raised to the fourth power.
To calculate the temperature of the heat lamp, we need to use both equations and solve for the unknown variable (temperature).
Here's the step-by-step process:
1. Convert the given wavelength from micrometers (μm) to meters (m). Since 1 micrometer equals 1 × 10^-6 meters, the wavelength becomes 6.7 × 10^-6 m.
2. Use Planck's law to relate the power (P) and the temperature (T) of the heat lamp at the given wavelength:
P = C × (λ^5) / (e^(D × λ × T) - 1)
At this point, we don't have enough information to directly solve for the temperature, as we don't know the constants C and D. However, we can simplify the equation by assuming that the exponential part (e^(D × λ × T) - 1) is very large compared to 1. This is typically the case for most practical scenarios.
By making this simplification, the equation becomes:
P ≈ C × (λ^5) / (e^(D × λ × T))
Keep in mind that this approximation is valid only if the exponential term is significantly larger than 1.
3. Rearrange the equation to isolate the temperature (T):
e^(D × λ × T) ≈ C × (λ^5) / P
4. Take the natural logarithm (ln) of both sides of the equation to remove the exponential and calculate the temperature:
D × λ × T ≈ ln(C × (λ^5) / P)
5. Solve for the temperature (T):
T ≈ ln(C × (λ^5) / P) / (D × λ)
Since we don't have the values for constants C and D in the question, we can't obtain the exact temperature from 20.0 watts of power at a wavelength of 6.7 μm without further information.
However, using the above steps, you can calculate the temperature if you have the values of the missing constants.
To calculate the energy of the heat lamp, we will need to use the formula:
Energy = Power × Time
Given:
Power = 20.0 watts
Wavelength = 6.7 μm (micrometers)
To find the energy, we need to convert the wavelength from micrometers to meters since the power is given in watts:
6.7 μm = 6.7 × 10^(-6) m
Now, we can calculate the energy using the formula mentioned above. However, we need to know the time for which the heat lamp is operating. Without this information, we cannot provide a specific answer. If you could provide the time, we can calculate the energy for you.