Electric Circuits Legacy Problem #34 Guided Solution

Electrical devices within your kitchen are typically wired up in a parallel circuit such that each of the devices is in its own separate branch. This allows you to turn the device off without shutting down the entire circuit. That would be the case for a series circuit. So one of the disadvantages of having all of your devices wired up in parallel is that as you turn more and more and more devices on, the overall resistance of the circuit tends to decrease and the current within that circuit tends to increase. This leads to potential fire hazards as the current begins to heat up the wires and the wires could catch on fire behind the framing of your house. Now, in order to protect such circuits against this problem, we put fuses or circuit breakers within the circuit that shut down when the current exceeds a certain level. For instance, a 20 ampere circuit breaker will shut down a circuit as soon as the current within that circuit exceeds 20 amperes. In this problem, we have a question about a culinary household who begins to do some cooking as they often do and begins to wire up or turn on different devices within the circuit. And we wish to find the overall current within the circuit as more and more and more devices are turned on. So there's a progression here as we go from A through Part D where more devices are being turned on and we begin to notice that the current within the circuit begins to increase and we have to calculate that current. Eventually, they ask us at what point on the turning on of these devices does the circuit become overloaded and the circuit breaker trips. Now one of the big concepts useful for analyzing parallel circuits like this complex one is to recognize that as charge travels out of the voltage source and through the wires, it comes to a point where it begins to branch and travel through a single branch. So a charge traversing the entire loop of a parallel circuit only goes through a single resistor. As it passes through the power source, it actually gains voltage and as it passes through the resistor or electrical appliance, it loses voltage. But the voltage gained equals the voltage lost and so the delta V or electric potential difference impressed across each one of these devices is 110 volts. So if you happen to know the resistance of the device, you could simply go 110 volts divided by that resistance and that gives you the current through that device. The other big principle for analyzing parallel circuits like this which will be useful in this problem is to recognize that the current outside the branches is simply the sum of the individual current values within the devices. So as you go through this collection or series of problems, what you want to do is calculate the current through each of the individual devices, through the mixer and the crockpot, through the mixer, the crockpot and the juicer. Do it independently and then sum them up. Now just because you add another device to your circuit, doesn't change the current through the other devices. The current through the mixer is still going to be the 110 volts divided by the 81 ohms regardless of whether the crockpot and the juicer are on or not. And so I would organize yourself here. Calculate the current through the mixer, calculate the current through the crockpot, it can be done independently. It's just simply the 110 volts divided by the resistance of that device. Write this stuff down and then as you go through part A through part G, begin to sum them up in order to get the total current for whatever devices are on. So let me just take part C as an example here and we'll calculate the current through the mixer, the current through the crockpot, through the juicer and the blender and the overall current. It would go something like this. The mixer has a resistance of 81 ohms. So for the mixer we can find the current through it as 110 volts divided by 81 ohms. That comes out to be 1.3580 amps. The crockpot has a resistance of 62 ohms. So to find the current in it we can go 110 volts divided by 62 ohms and we get 1.7742 amps. The juicer has a resistance of 43 ohms. So to find the resistance of it we can go 110 volts divided by 43 ohms and we get 2.5581 amps. The blender has a resistance of 21 ohms. So 110 volts divided by 21 ohms is 5.2381 amps. If we sum these four different amperages, or current values, we would get the total current in the circuit when all four of these devices are on. And that sums up to be 10.9284 amps. That's the answer to part C. And as you proceed to part D, you would keep this number 10.9284 amps and simply add onto it 110 volts divided by the resistance of the next device. Continue the process and you should get the answers that are shown here on this page.