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
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.