Video Transcript
A wire carries a current of two plus or minus 0.1 amperes for a time of 40 plus or minus 0.5 seconds. What is the percent uncertainty of the charge that passes through the wire in that time?
Here then, we have a wire carrying a current, we’ll call 𝐼, for an amount of time, we’ll call 𝑡. Our question asks us to solve for the percent uncertainty of the charge that passes through the wire in this time 𝑡. In general, current 𝐼 is equal to the amount of charge 𝑄 that passes a point over an amount of time 𝑡. We can see then that if we multiply both sides of this equation by the time 𝑡 so that it cancels on the right, the total charge 𝑄 passing a given point in a wire equals the current in the wire 𝐼 times the time elapsed 𝑡. In our scenario then, 𝑄 equals 𝐼 times 𝑡 or two plus or minus 0.1 amperes multiplied by 40 plus or minus 0.5 seconds.
Before we’re able to solve for the percent uncertainty in this charge 𝑄, we’ll want to solve simply for the uncertainty in 𝑄. We can see that 𝑄 will be equal to two times 40 or 80 ampere seconds. But there is an uncertainty in 𝑄 as well. To figure out what that uncertainty is, let’s imagine we have two numbers 𝑎 and 𝑏, each with its own uncertainty 𝜎𝑎 and 𝜎𝑏. If 𝑐 is equal to the product of 𝑎 and 𝑏, then the uncertainty in that product, we’ll call it 𝜎𝑐, equals 𝑏 times 𝜎𝑎 plus 𝑎 times 𝜎𝑏.
In our equation for 𝑄, we can think of the two as 𝑎 and 0.1 as 𝜎𝑎 and then 40 as 𝑏 and 0.5 as the uncertainty in 𝑏. Therefore, the uncertainty in 𝑄 is 𝑏, that’s 40, times the uncertainty in 𝑎, 0.1, plus 𝑎, that’s two, times the uncertainty in 𝑏, that’s 0.5. And as we’ve seen, the units of 𝑄 are ampere seconds. The unit ampere times seconds by the way is equal to coulombs. So the total charge 𝑄 equals 80 plus or minus this quantity 40 times 0.1, that’s four, plus two times 0.5, that’s one. So when we add four and one, we find that the total uncertainty in 𝑄 is five coulombs of charge.
Note that this isn’t quite our final answer because we want to solve for the percent uncertainty in 𝑄. To find the percent uncertainty in some number, say we’re working with the number 𝑎 plus or minus 𝜎 sub 𝑎, we take the uncertainty divided by the number itself and then multiply that fraction by 100 percent. That will give us the percent uncertainty in 𝑎. This means then that the percent uncertainty in 𝑄 is equal to the uncertainty of 𝑄, that’s five divided by 𝑄 which is 80. Note that we’re leaving out the units of coulombs all multiplied by 100 percent. When we calculate this expression, we get exactly 6.25 percent. This is the percent uncertainty of the charge that passes through the wire in the given amount of time.
