Ohm's Law and Battery Life

Date: 08/18/99 at 17:04:12
From: Scott Stongle
Subject: Ohm's law

If you have a 12 V battery that has 275 cold-cranking amps and you put 
a load of 12 V at 7.2 A on it, how long should that battery run the 
load? Also, I was wondering if I need to figure the resistance of the 
system before I can complete the calculation.

Thank you.

Date: 08/23/99 at 11:04:20
From: Doctor Douglas
Subject: Re: Ohm's law

Hi Scott -

Thanks for writing. Let's see if the following helps.

Using Ohm's law, Voltage = Current * Resistance:

     12 V = 7.2 A * R

So the resistance of the load is 12 V/7.2 A = 1.67 Ohms. We've just 
figured out the resistance of the load. There is also an internal 
resistance of the battery that limits how much current it can supply, 
even to a short circuit. This limit is the 275 amps (the maximum 
current that can be drawn). So the internal resistance of the battery 
is given by:

     12 V = 275 A * [R_battery + R_short]
          = 275 A * [R_battery + 0]
          = 275 A * R_battery

Now we have R_battery = 12 V/275 A = 0.0436 Ohm, which is small 
indeed. If the load is smaller than about 10*R_battery, then the 
battery will not have enough oomph to put out 12 V into a load that 
small.

Now since our load (1.67 Ohms) is bigger than R_battery, there's no 
problem so far - the battery will put 12 V across the load and be able 
to deliver the 7.2 amps, which is much less than its maximum rating of 
275 amps.

Now let's see if there's enough information to answer your question, 
"how long will it run the load?" The property of the battery that you 
need to know here is known as its "capacity," and this expression is 
usually given in units of ampere-hours. This capacity is illustrated 
by the difference in battery sizes: a D-cell (flashlight) and a AA 
cell (Walkman) have the same voltage (1.5 V), and probably very close 
to the same R_battery, but the D-cell has much higher capacity (and 
hence is bigger and heavier, because it can last longer) than the AA. 
As you might imagine, if the load is smaller it will require more 
current and the lifetime of the battery will be smaller.

In your question, we don't have this information on the battery 
capacity, so at this point we can't yet answer how long it will last. 
If you find out the battery capacity, then you can calculate the 
lifetime from 

     lifetime = (capacity) / (load-current)

assuming that everything remains constant. Of course, real batteries 
and real loads might behave slightly differently, and this could make 
things a bit more complicated.

Hope this helps! If you are still unclear about anything, please write 
back.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/