A 420-volt AC supply is short-circuited by a conductor of resistance of 0.850 ohms. How much power is dissipated in the short circuit? What is the current in the conductor?
First, let’s consider what it means that this AC supply is short-circuited. First, let’s imagine this power supply in a circuit with some resistive load. We’ll just call it 𝑅 sub load. We don’t know what that resistance is. But this is the circuit as it was designed, we’ll say. When a circuit like this is short-circuited, that means another path for the current develops that has much less resistance than the native path in the original circle.
In this case, we imagine that our short-circuit path has a very small resistance value of 0.850 ohms. This resistance is so small that, almost regardless of the value of 𝑅 sub load, we don’t know what it is. But we can assume it’s much larger that the current, which moved through this circuit and previously followed the path through the load resistor, will now instead prefer this much lower resistant path. It’s short-circuited.
The first question then is, with this same power supply and this new resistor value, what is the power dissipated in this short circuit? One way to express power in electrical circuits is that it’s the voltage in the circuit squared divided by the resistance of the circuit. In our case, we can compute the power by dividing the voltage, 420 volts, by the resistance of the short circuit branch, 0.850 ohms. This is equal to 208000 watts or 208 kilowatts. That’s the power dissipated in the short circuit.
In our next step, we want to solve for the current that runs through this conductor, that is, through this branch of the short circuit. Looking at our short circuit diagram, we see we have an overall power supply, 420 volts, and an overall resistance. So we want to connect those two values with current in the conductor. We make that connection through Ohm’s law, which says that the potential difference is equal to the product of current and resistance.
Rearranging that equation to solve for our variable of interest, current, we find that it’s equal to the voltage 𝑉 divided by the resistance 𝑅. That’s equal to 420 volts divided by 0.850 ohms or 494 amps. That’s how much current flows through the conductor in the short circuit.