Question Video: Calculating Number of Quanta Emitted by a Blackbody Physics

Video Transcript

The total energy emitted by a black body is 1.25 microjoules. If the total radiated energy is divided into quanta with a wavelength of 700 nanometers, how many such quanta would be emitted? Use 6.626 times 10 to the power of negative 34 joule-seconds for the value of Planck’s constant. Answer in scientific notation to two decimal places.

In this question, we are asked to calculate the number of quanta emitted by this black body. We’re told in the question that the total energy emitted, 1.25 microjoules is divided into quanta which all have wavelengths of 700 nanometers. Recall that the energy of one of these quanta of light, 𝐸 q, is given by the equation 𝐸 q equals ℎ𝑐 over 𝜆, where ℎ is Planck’s constant, 𝑐 is the speed of light, and 𝜆 is the wavelength of the quanta.

So if we have 𝑛 quanta of light emitted by the black body, the total energy 𝐸 will be given by 𝐸 equals 𝑛ℎ𝑐 over 𝜆. To find our answer, 𝑛, we need to rearrange this equation to make 𝑛 the subject. We start by multiplying both sides of the equation by the wavelength, which cancels out the wavelength on the right-hand side. Then by dividing both sides by ℎ𝑐, we find that the number of quanta 𝑛 is equal to 𝐸𝜆 over ℎ𝑐.

We are now in a position to substitute in our values. Total energy is given at 1.25 microjoules, which is equal to 1.25 times 10 to the power of negative six joules. Wavelength is given as 700 nanometers, which is equal to 700 times 10 to the power of negative nine meters. We are given the value of Planck’s constant as 6.626 times 10 to the power of negative 34 joule-seconds. And we can recall that the speed of light is equal to three times 10 to the power of eight meters per second.

Substituting in these values into the equation and completing the calculation, we find that 𝑛 is equal to 4.4018 times 10 to the power of 12, which has no units as it is just a number. To two decimal places, we find that 𝑛 is equal to 4.40 times 10 to the power of 12. And so this is our final answer. If the total radiated energy is divided into quanta with a wavelength of 700 nanometers, then the number of quanta emitted will be equal to 4.40 times 10 to the power of 12.