How do I work out an expression for the total energy

• bon
In summary, the function n(E) describes the number of photons with energy between E and E+dE. There is no upper limit on E (although n(E) -> 0 as E -> 0). The energy of all photons with energy between E and E+dE is proportional to T^3.
bon

Homework Statement

Black body radiation inside a cavity at temperature T may be thought of as a gas of photons with an energy distribution given by the function n(E) given by

n(E)dE = A [E^2 / (e^(E/kT) - 1) ] dE

Where A is independent of E and T. (k is Boltzmann constant)

The function n(E) describes the number of photons with energy between E and E + dE. There is no upper limit on E (although n(E) -> 0 as E -> 0)

Show that

a) the total number of photons is proportional to T^3 and
b) the total energy is proportional to T^4

The Attempt at a Solution

So I'm a bit confused!

To work out the total number of photons, do I just do the integral of n(E)dE with E from 0 to infinity? How is this proportional to T^3?

How do I work out an expression for the total energy (since n(E) just describes the number of the photons..) ?

Thanks

bon said:

To work out the total number of photons, do I just do the integral of n(E)dE with E from 0 to infinity? How is this proportional to T^3?

You don't actually have to do the integral to show the proportionality. Change variables to

$$x = E/kT$$

and you can factor the T dependence out of the integral completely.

How do I work out an expression for the total energy (since n(E) just describes the number of the photons..) ?

If n(E) is the total energy of the photons with energy between E and E+dE, can you write an expression for the energy of all photons with energy between E and E+dE?

If you can't, suppose there were 4 photons with energy between E and E+dE. What is the total energy of these 4 photons?

Ah great thanks! Ok so is it okay to just say: integral from 0 to infintiy of n(E) dE = A k^3 T^3 times integral from 0 to infnity of x^2 / e^x -1 dx? (because i can't actually see any way of solving the integral..)

Oh is the energy just nE? so i integrate En(E) between 0 and infinity..hence the extra T power? yay i think i see

great help thanks//

What is total energy?

Total energy refers to the sum of all forms of energy present in a system, including kinetic, potential, thermal, and chemical energy.

Why is it important to work out an expression for total energy?

Working out an expression for total energy allows us to understand and calculate the amount of energy in a system. This knowledge is crucial in many areas of science, such as thermodynamics, mechanics, and chemistry.

How do I determine the total energy of a system?

The total energy of a system can be determined by adding together the energies of all individual components, taking into account any conversions or transformations that may occur.

What factors affect the total energy of a system?

The total energy of a system can be affected by various factors such as the mass of the objects involved, their velocities, the distance between them, and any external forces or fields present.

Can the total energy of a system change?

According to the law of conservation of energy, the total energy of a closed system remains constant. However, the energy within the system can change form or be transferred to or from the surroundings.

Replies
4
Views
1K
Replies
0
Views
718
Replies
6
Views
1K
Replies
4
Views
853
Replies
9
Views
850
Replies
7
Views
2K
Replies
1
Views
1K
Replies
8
Views
1K