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When I try to do it I get p=ρ/6. I was hoping somebody could tell me where I'm going wrong.

Here is my working.

Say all photons have frequency ##\nu## and the number of photons per cubic metre is ##n##. Then ##\rho=n h \nu##.

Now consider one face of a cube of side length 1m. The pressure on that face is the component, parallel to the normal to the face, of the impulse delivered to that face in one second, by photons striking it from outside the cube. We can ignore components of impulse parallel to the face, because the isotropy will make them cancel each other out.

We measure that impulse by integrating over all photons that at time 0 are in a hemisphere of radius c from the centre of that face, with the base of the hemisphere coplanar with the face. No photons outside the hemisphere can strike that face in the next second.

We use spherical coordinates for the hemisphere and consider a volume element ##dV## from ##r## to ##r+dr##, ##\theta## to ##\theta+d\theta##, ##\phi## to ##\phi+d\phi##

The solid angle subtended by the cube's face at the centre of ##dV## is ##\frac{cos\theta}{r^2}##. Hence, since the gas is isotropic, the proportion of photons in ##dV## that will strike that face is ##\frac{cos\theta}{4\pi r^2}## and the number of photons in ##dV## at time 0 that will strike the face is ##\frac{n dV cos\theta}{4\pi r^2}##. The impulse delivered by those photons parallel to the face's normal will be ##\frac{h\nu}{c}cos\theta\frac{n dV cos\theta}{4\pi r^2}## .

Hence the pressure on the face is ##p = \int\frac{h\nu}{c}cos\theta\frac{n cos\theta}{4\pi r^2}dV

= \int_0^c\int_0^\frac{\pi}{2}\int_0^{2\pi}\frac{ h\nu}{c}cos\theta\frac{n cos\theta}{4\pi r^2}dr (r d\theta) (r sin\theta\ d\phi)##

##

= \int_0^c\int_0^\frac{\pi}{2}\int_0^{2\pi}\frac{n h\nu}{4\pi c}cos^2\theta sin\theta\ dr d\theta d\phi##

## = 2\pi c\frac{n h\nu}{4\pi c}[-\frac{1}{3}cos^3\theta]_0^\frac{\pi}{2}

= \frac{n h\nu}{6}##

Now the energy density is ##\rho=n h\nu## so we have p=ρ/6 rather than the desired p=ρ/3.

What have I done wrong?

Thank you in advance for all suggestions.