Movie Energy

Hollywood sells some of the most expensive energy on earth.

You go to the theater, pay $8, and absorb 120 minutes of light and sound energy into your pupils and ears.

How much light energy?

The average dark-adapted pupil size is 7mm. Movie theaters at maximum brightness (no film in the projector gate) are recommended to produce 16 foot-lamberts of light. A foot-lambert is 3.426 candela/m². A candela is 1/683 watts per steradian. Assume the screen is a Lambertian reflector with a gain of 1 and an aspect ratio of 2:1 , and you are sitting one screen width away.

A white "pixel" at screen position (x,y) of size (dx,dy) radiates 3.426*16 = 54.8 cd *dx*dy of light, or P_0*dx*dy = 80mW/steradian *dx*dy . A Lambertian reflector at distance R, and off axis by angle θ delivers light with an intensity of (A*dx*dy)*cos(θ)/π R² W/m. cos(θ)² = x²+y² / R² . If z = W is the distance from the screen, then R² = x² + y² + W² . The total light power received by a surface at that distance is this integral:

P ~ = ~ \int_{-W/2}^{+W/2} \int_{-W/4}^{+W/4} P_0 \cos( \theta ) / R^2 dx dy

P ~ = ~ \int_{-W/2}^{+W/2} \int_{-W/4}^{+W/4} P_0/\pi \sqrt{ (x^2+y^2)/ (x^2+y^2+ W^2) } / (x^2+y^2+ W^2) dx dy

let x' = x / W and y' = y/W , this becomes:

P ~ = ~ P_0/\pi \int_{-1/2}^{+1/2} \int_{-1/4}^{+1/4} \sqrt{ (x'^2+y'^2)/(x'^2+y'^2+ 1)^3 } dx' dy'

There is probably an analytical expression for that integral, but I am lazy and computed it numerically instead, ≈ 0.12348 , so P ≈ 3 mW/m², or 3 nW/mm².

A white screen produces about 120 nW in a 7mm diameter pupil, and an "average" screen might produce 60 nW at the retina. Integrated over 7200 seconds, that is about 400 μJoules.

How much sound energy?

Let's assume you only go to loud PG-13 action movies, because you want to harvest the maximum amount of sound energy (english drawing room mysteries are a much worse value!). Assume 0dB = 1e-12W/m²; the sound power increases by a factor of 10 for each 10dB. At 131.8 dB, or 15W/m², with a 7mm diameter ear canal, your ear gets about 600 microwatts of sound power. This is 10,000 times more power than your eye gets, about 4 Joules per movie.

So, your movie-going energy-absorption experience is almost entirely sound. Therefore, blind people can get just as much energy value as the sighted, and movie companies can dispense with sets, cameras, cinematographers, and pretty actors. Just subtitles for the hard of hearing, which will be much easier to see without distracting moving pictures.

What a value!

How does that compare to 12 cents per kilowatt-hour (US average, ranging from 7.5 cents to 30 cents) electricity from the power company?

\$8 per 4 Joules = \$7.2M per kwh , or 60 million times more expensive.

While this seems like a bad bargain (you could just sit at home, wired into a wall socket), keep in mind that many Hollywood actors are fond of alternative energy sources, some of which are outrageously expensive. If we do not keep these people busy in front of cameras, they will be busy pestering Congress for more Solyndra subsidies.

Until we find a safe way to sequester glamorous non-scientists away from our leaders, it is our patriotic duty to go to movies and keep them distracted!