Question Video: Calculating Currents and Voltages in a Parallel Circuit

The total current in the circuit shown is 100 mA. a) What is the total resistance in this circuit to the nearest ohm? b) What is the voltage across the 40 Ω resistor to 2 significant figures? c) What is the current in the 50 Ω resistor to the nearest milliampere?

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Video Transcript

The total current in the circuit shown is 100 milliamperes. What is the total resistance in the circuit to the nearest ohm?

The resistors in the circuit are in parallel because there are two or more paths for the current to take. So before we find the total resistance of the circuit, let’s review how resistors behave when they’re in parallel. Here, I’ve drawn a circuit with two resistors in parallel, a voltage 𝑉, and a current 𝐼.

To find the total resistance of the resistors in parallel, we would sum the inverses of the resistances for each resistor and then take the inverse of that. So in this circuit, with a 30, 40, and 50 ohm resistor, we can find the total resistance by summing one over 30 plus one over 40 plus one over 50. This gives us 0.0783 repeating inverse ohms, which is equal to one over the total resistance. So, the total resistance will be equal to one over this number. Which gives us 12.77 ohms. So, the total resistance of the circuit to the nearest ohm is 13 ohms.

What is the voltage across the 40 ohm resistor to two significant figures?

Before we answer this part of the question, let’s review how both voltage and current behave across resistors that are in parallel. In a circuit in parallel, the voltages dropped across each branch of the circuit will be the same. The current in a parallel circuit will be divided among each branch of the circuit. And, the current of each branch will sum to the total current of the circuit. We want to find the voltage across the 40 ohm resistor in the circuit. But, because all of the resistors in the circuit are in parallel, the voltage across each of them will be the same. We can use the total resistance that we found in the previous part of the problem to draw a circuit that’s equivalent to this one.

The voltage across the 13 ohm resistor in this circuit will be equivalent to the voltage across the 40 ohm resistor or, in fact, any of the resistors in the other circuit. We can now use Ohm’s law to solve for the voltage. The total current in our circuit is 100 milliamps, and the total resistance is 13 ohms. Before we can solve for the voltage, we need to first convert our current from units of milliamps to units of amps. Since there are 1000 milliamps in an amp, we can convert from milliamps to amps by dividing by 1000. Now, we can solve for the voltage, which is 1.30 volts. Rounding to two significant figures, the total voltage in our circuit, and therefore the voltage across the 40 ohm resistor, is 1.3 volts.

What is the current in the 50 ohm resistor to the nearest milliampere?

We’re going to use Ohm’s law again to find the current across the 50 ohm resistor. The current will be equal to the voltage divided by the resistance. The voltage across this resistor is the same as the voltage that we just found in the previous part of the problem, 1.30 volts. The resistance is 50 ohm. This gives us 0.0260 amps. The question asked us to express our answer to the nearest milliampere. So, we can convert to milliamperes by multiplying by 1000 because there’s 1000 milliamperes in an ampere. This gives us 26.0 milliamperes. So, the current in the 50 ohm resistor to the nearest milliamp is 26 milliamps.

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