Resistor Circuit Analysis: Calculate Resistance, Current, Voltage, And Power

by TextBrain Team 77 views

Alright, physics enthusiasts! Let's dive into a classic circuit problem involving resistors. We've got a circuit with a 12V source and a bunch of resistors: 1Ω, 3Ω, 2Ω, and another 3Ω. Our mission, should we choose to accept it, is to calculate the total resistance, the current through each resistor, the voltage across each resistor, and the power dissipated by each resistor. Buckle up, because we're about to embark on an electrifying journey!

a) Calculating the Total Resistance

So, first things first, let's figure out the total resistance of this circuit. To do this, we need to know how these resistors are connected. Are they in series, parallel, or a combination of both? Let's assume for a moment that the 3Ω, 2Ω, and 3Ω resistors are in series with each other, and then that whole series combination is in parallel with the 1Ω resistor. We are also assuming that the 12V source is connected to the entire resistor network.

If the resistors are connected in series, the total resistance is simply the sum of the individual resistances. If R1, R2, and R3 are in series, then the equivalent resistance (Req) is:

Req = R1 + R2 + R3

If the resistors are connected in parallel, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. If R1, R2, and R3 are in parallel, then the equivalent resistance (Req) is:

1/Req = 1/R1 + 1/R2 + 1/R3

Given our assumption, let's calculate. The series combination of the 3Ω, 2Ω, and 3Ω resistors is:

Rs = 3Ω + 2Ω + 3Ω = 8Ω

Now, this 8Ω resistor is in parallel with the 1Ω resistor. To find the total resistance of this parallel combination, we use the formula:

1/Rp = 1/1Ω + 1/8Ω

To solve for Rp, we first find a common denominator:

1/Rp = 8/8Ω + 1/8Ω = 9/8Ω

Now, we take the reciprocal of both sides to find Rp:

Rp = 8/9 Ω ≈ 0.89 Ω

So, the total resistance of the circuit, with our assumed configuration, is approximately 0.89Ω. Remember, the actual total resistance depends heavily on how the resistors are wired together!

b) Calculating the Current Through Each Resistor

Alright, now that we've wrestled with the total resistance, let's move on to calculating the current through each resistor. To do this, we'll need to use Ohm's Law, which states:

V = IR

Where:

  • V is the voltage across the resistor
  • I is the current through the resistor
  • R is the resistance of the resistor

To find the current (I), we can rearrange Ohm's Law to:

I = V/R

Let's start with the 1Ω resistor. Since it's in parallel with the series combination of the other resistors, it experiences the full 12V from the source. Therefore, the current through the 1Ω resistor is:

I1 = 12V / 1Ω = 12A

Now, for the series combination of the 3Ω, 2Ω, and 3Ω resistors, we first need to find the voltage across this combination. Since the 1Ω resistor is in parallel, the voltage across the series combination is also 12V. The current through this series combination is the total current flowing through the equivalent 8Ω resistance:

Is = 12V / 8Ω = 1.5A

Since the resistors in a series circuit have the same current flowing through them, the current through each of the 3Ω, 2Ω, and 3Ω resistors is 1.5A.

So, to recap:

  • Current through the 1Ω resistor: 12A
  • Current through each of the 3Ω, 2Ω, and 3Ω resistors: 1.5A

c) Calculating the Voltage Across Each Resistor

Next up, let's calculate the voltage across each resistor. We'll be using Ohm's Law again, but this time, we'll solve for V:

V = IR

For the 1Ω resistor, we already know the voltage: it's 12V, since it's directly connected to the voltage source. But let's double-check using Ohm's Law:

V1 = 12A * 1Ω = 12V

Now, let's find the voltage across each of the series resistors. Remember, the current through each of these resistors is 1.5A.

  • Voltage across the first 3Ω resistor: V3_1 = 1.5A * 3Ω = 4.5V
  • Voltage across the 2Ω resistor: V2 = 1.5A * 2Ω = 3V
  • Voltage across the second 3Ω resistor: V3_2 = 1.5A * 3Ω = 4.5V

Notice that the sum of the voltages across the series resistors (4.5V + 3V + 4.5V) equals 12V, which is the total voltage across the series combination. This is a good way to check your work and make sure you haven't made any mistakes!

d) Calculating the Power Dissipated by Each Resistor

Last but not least, let's calculate the power dissipated by each resistor. The power dissipated by a resistor is the rate at which electrical energy is converted into heat. We can calculate power using the following formula:

P = IV

Where:

  • P is the power in watts
  • I is the current in amperes
  • V is the voltage in volts

We can also express power in terms of resistance using Ohm's Law. Substituting V = IR into the power equation, we get:

P = I^2 * R

Or, substituting I = V/R into the power equation, we get:

P = V^2 / R

Let's calculate the power dissipated by each resistor using P = IV:

  • Power dissipated by the 1Ω resistor: P1 = 12A * 12V = 144W
  • Power dissipated by the first 3Ω resistor: P3_1 = 1.5A * 4.5V = 6.75W
  • Power dissipated by the 2Ω resistor: P2 = 1.5A * 3V = 4.5W
  • Power dissipated by the second 3Ω resistor: P3_2 = 1.5A * 4.5V = 6.75W

So, there you have it! We've calculated the power dissipated by each resistor in the circuit. Notice that the 1Ω resistor dissipates significantly more power than the other resistors. This is because it has a much larger current flowing through it.

Conclusion

Phew, that was quite the workout! We successfully calculated the total resistance, the current through each resistor, the voltage across each resistor, and the power dissipated by each resistor. Remember, the key to solving circuit problems is to understand the relationships between voltage, current, and resistance, and to apply Ohm's Law and the power equation correctly. And of course, always double-check your work to make sure you haven't made any silly mistakes. Keep practicing, and you'll become a circuit-solving pro in no time! Keep in mind that these calculations were based on the assumption of how the resistors are connected. If the configuration is different, you'll need to adjust your calculations accordingly. Always visualize the circuit and understand the flow of current before diving into the math!