Lighting the Way

Headlights: evil harbingers of environmental doom.

Lately, I've seen a lot of cars driving with their headlights on in the day time.   This seems like a waste of energy.  Aren't we supposed to turns the lights off when we're not using them?  Not only does it cost energy to produce the light, but the lights will actually exert a tiny force backwards on your car.  In principle, this will slow your car's acceleration by a tiny amount and cause you to use more gas.  How much gas do you waste by turning the on the headlights?

Most car lights look yellow.  Yellow light has a wavelength and frequency of λ = 570 nm and f = 5×10¹⁴ Hz, respectively.  Each photon of light that leaves the car carries with it some energy given by¹

E = h f .

If you have N photons emitted by the car, the car will have lost an amount of energy

ΔE = N h f .

Car lights consume energy at a rate of about P = 50 W.  This P is known as power, and it's the rate at which energy is transferred,

P = ΔE / Δt.

Assuming all the energy goes into creating yellow photons, we can find how many of particles of light leave the car each second by solving for N:

N = P Δt / h f, 

= (50 W) · (1 sec) / (6.63×10^-34 J · s) · (5×10¹⁴ Hz),

= 1.5×10²⁰ photons.

Each photon that leaves carries with it some momentum,

p = h / λ.  

As it leaves, the photon imparts some force on the car,

F = Δp / Δt.

The total force on the car is then

F = N h / λ Δt ,

= (1.5×10²⁰ photons) · (6.63×10^-34 J · s per photon) / (540 nm) · (1 sec),

= 1.8×10^-7 N.

That's a tiny force.  Just how tiny?  If you drive 3000 miles across the United States, it'll cost only 1 Joule of energy.²  One gallon of gasoline contains 1.3×10⁸ J of energy.  You would need to drive around the U.S. with your lights on 130 million times before you'd wasted a gallon of gas.

[1] Photons are particles of light.
[2] You can obtain this by multiplying the force times the distance.