Question

Can you please help me to solve this exercise? I know it is quite long but please try to do as much as your are allowed to do.

Thank you

5.- (Pressure and energy density of electromagnetic radiation) Consider the electromagne- tic radiation (photon gas) contained in an edge box Lg, Ly and L2. As an image moves with the speed of light, it is a relativistic particle. Therefore, its energy e is related to its moment hK through: к3)12 кз. € — сhK — сh(К- (2) (a) Use this relation to calculate the force Fr exerted by a photon vessel when the photon is in a state r specified by nx, ny, nz. on the right wall of the (b) Using the average energy of a photon. Use the symmetry requirement K2 K2 the radiation is in equilibrium with the walls of the enclosure expression for the average force F as a function of simple average, deduce an a K2, when (c) From here show that the average pressure p exerted on the walls by radiation is given by: 1 (3 where uis the electromagnetic energy of the radiation per unit volume.

Answer 1

elementof = chk = ch (k_x^2 + k_y^2 + k_z^2)^1/2 = ch squareroot (n_x pi/L_x)^2 + (n_y pi/L_y)^2 + (n_z pi/L_z)^2 Therefore so the force F_x exerted by proton is F_x = -Partialdifferential elementof_y/Partialdifferential L_x = -ch Partialdifferential/Partialdifferential L_x [(n_x pi/L_x)^2 + (n_y pi/L_y)^2 + (n_z pi/L_z)^2]^1/2 = -ch -1 middot 2 middot (n_x pi/L_x) middot n_x pi/L_x^2/2 squareroot (n_x pi/L_x)^2 + (n_y pi/L_y)^2 + (n_z pi/L_z)^2 = ch (n_x pi)^2 1/L_x^3/squareroot (n_x pi/L_x)^2 + (n_y pi/L_y)^2 + (n_z pi/L_z)^2 = ch (n_x pi)^2 1/L_x^3/squareroot (n_x pi/L_x)^2 + (n_y pi/L_y)^2 + (n_z pi/L_z)^2 \r\n F = Sigma_r P_r F_r = Sigma e^-beta elementof_r (-Partialdifferential elementof_r/PartialdifferentialL_x)/Sigma_r e^-beta elementof_r = 1/beta Partialdifferential ln z/Partialdifferential L_x Now, z = e^-beta ch squareroot (n_x pi/L_x)^2 + (n_y pi/L_y)^2 + (x_z pi/L_z)^2 ln z = -beta ch [(n_x pi/L_x)^2 + (n_x pi/L_y)^2 + (x_2 pi/L_z)^2]^1/2 now using symmetric argument k_x^-2 = K_y^-2 = K_z^2 ln z = -beta ch