Just how expensive are batteries?

Here is an interesting little physics calculation. How does the cost of batteries compare to the cost of "wall" electricity?

Well, the cost you pay your electric company will be around 15 cents per kilowatt-hour. (The non-dimensionally challenged will note that you don't pay for power, which would be in kilowatts, but energy, which is in kilowatt hours.)

For a battery, let's take:

• Voltage, V = 1.5 volts
  
• Average Current, I = 2/3 amps
  
• Lifetime, T = 5 hours
  
• Cost, C = 1 dollar = 100 cents
  

These numbers are representative of a D cell battery.

Let's get the correct formula by dimensional analysis—this means if we arrange the parameters to give us the correct dimensions then, we have be close to the correct formula.(That is, there is a unique arrangement of the relevant parameters that produce the correct units.)

Our answer should be in cents per kilowatt hour, which means cents/kilowatt-hour

To get watts we need volts × amps, or V × I. To get kilowatts, we need V × I / 1000. Thus to get cents per kilowatt hour we need

(C × 1000) / ( V × I × T)

To be sure, dimensional analysis only tells us the answer is "like" this. There could be a factor of 2 or π that we missed. This, however, turns out to be the correct formula, one that we would have arrived at by more traditional reasoning.

So what do we get? Plug in the numbers:

(100 cents × 1000) / (1.5 volts × 2/3 amps × 5 hours) = 20,000 cents per kilowatt-hour! (or 200 dollars per kilowatt-hour)

Thus a battery is ~1300 times more expensive that wall electricity.

For more "back of the envelope" calculations like this one, see Back of the Envelope Physics, by Clifford Swartz.