The shortest visible wavelength is about 400 nm. What is the temperature of an ideal radiator whose spectral emittance peaks at this wavelength?

Video transcript

Hey, everyone. Let's go through this problem. A hot plate emits a continuous spectrum and has its peak intensity wavelength at 610 nanometers calculate the absolute temperature of the hot plate. Assuming that it acts as a black body, we have four options to choose from. Option A 1.77 multiplied by 10 to the power of three Kelvins. Option B 2.10 multiplied by 10 to the power of three Kelvins. Option C 2.90 multiplied by 10 to the power of three Kelvins. And option D 4.75 multiplied by 10 to the power of three calvins. So because we're looking for the temperature and we're given the wavelength at which the spectrum has its peak intensity, this is a perfect opportunity to use the beams law which states that the wavelength of maximum intensity is equal to the vein constant, 2.90 multiplied by 10 to the power of negative three m. Kelvins divided by the temperature of that spectrum. Now, because we're looking for the temperature, we'll first have to algebraically solve the INS law for T. So we can multiply both sides of the equation by T and then divide both sides of the equation by Lambda Max. And we find that the temperature is equal to the V constant 2.90 multiplied by 10 to the power of negative three m Kelvins divided by Lambda sub max. Now we're given the wavelength so we're given Lambda submax. So all we gotta do is plug 610 nanometers into this expression and solve for T. So what we find is that the temperature is equal to 2.90, multiplied by 10 to the power of a negative three m. Kelvins divide that by 610 nanometers. So that's 610. And then to convert this into meters, we multiply by 10 to the power of negative nine m. And when we plug this into our calculator, we find a temperature of about 4754. Kelvins over all of the options are given to us in scientific notation. So let's round this into scientific notation as 4.75 multiplied by 10 to the power of three Kelvins. And this is a temperature. So this is then the answer to our problem. And if we look at our options, we can see that this agrees with option B 4. multiplied by 10 to the power of three Kelvins. So option D is the answer to this problem and that's it for this video. I hope this video helped you out if it did. And if you're looking for more practice, please check out some of our other videos which will hopefully give you more experience with these types of problems, but that's all for now and I hope you all have a lovely day. Bye bye.