Series Circuit Analysis: Voltage, Resistance, Power

by Omar Yusuf 52 views

Hey guys! Today, we're diving into the fascinating world of series circuits. We're going to break down a circuit problem step-by-step, covering everything from drawing the schematic to calculating the power delivered by the source. So, buckle up and let's get started!

The Series Circuit Challenge

Our challenge involves a series circuit powered by a 120-volt DC ideal voltage source. This source is connected to six resistors with the following values:

  • R1 = 1250 Ω
  • R2 = 2120 Ω
  • R3 = 3330 Ω
  • R4 = 4650 Ω
  • R5 = 5330 Ω
  • R6 = 8330 Ω

Our mission, should we choose to accept it (and we do!), is twofold:

  1. Draw the circuit diagram. This helps us visualize the connections and understand the flow of current.
  2. Prove that the power delivered by the source is what we calculate. This involves using Ohm's Law and the power formula to verify our results.

Step 1: Drawing the Circuit Diagram

First things first, let's get visual. Drawing a circuit diagram is crucial for understanding how the components are connected. In a series circuit, components are connected one after the other, forming a single path for the current to flow.

Imagine a single lane road; all the cars (electrons, in this case) have to follow the same route. That's how a series circuit works!

Here’s how we represent our circuit diagram:

  1. Voltage Source: Draw a circle with a plus sign (+) on one side and a minus sign (-) on the other. This represents our 120V DC source. The longer line typically represents the positive terminal.
  2. Resistors: Draw a series of six rectangles, each representing a resistor. Label them R1, R2, R3, R4, R5, and R6, and write their respective resistance values (1250 Ω, 2120 Ω, 3330 Ω, 4650 Ω, 5330 Ω, and 8330 Ω) next to them.
  3. Connections: Connect the components in a single loop. Start from the positive terminal of the voltage source, connect it to R1, then connect R1 to R2, R2 to R3, and so on, until R6 is connected back to the negative terminal of the voltage source.

This diagram clearly shows how all the resistors are in a single line, connected sequentially, which is the hallmark of a series circuit. Visualizing the circuit makes understanding the next steps much easier.

Step 2: Calculating Total Resistance

In a series circuit, the total resistance (RT) is simply the sum of all individual resistances. This is because the current has to flow through each resistor in turn, encountering resistance at every step. It’s like adding up all the tolls on a long highway – the total toll is the sum of the individual tolls.

So, to find the total resistance in our circuit, we add up the values of all six resistors:

RT = R1 + R2 + R3 + R4 + R5 + R6

Plugging in the values:

RT = 1250 Ω + 2120 Ω + 3330 Ω + 4650 Ω + 5330 Ω + 8330 Ω

RT = 24,910 Ω

Therefore, the total resistance of our series circuit is 24,910 ohms. This value is crucial for calculating the current flowing through the circuit.

Step 3: Determining the Current Flow

Now that we know the total resistance and the voltage supplied by the source, we can calculate the current flowing through the circuit using Ohm's Law. Ohm's Law is a fundamental principle in electronics, stating the relationship between voltage (V), current (I), and resistance (R). It's expressed as:

V = I * R

We can rearrange this formula to solve for current (I):

I = V / R

In our case:

  • V (Voltage) = 120 V
  • RT (Total Resistance) = 24,910 Ω

Plugging these values into the formula:

I = 120 V / 24,910 Ω

I ≈ 0.00482 A

So, the current flowing through the circuit is approximately 0.00482 amperes (A), or 4.82 milliamperes (mA). Remember, in a series circuit, the current is the same at every point in the circuit. This means that 4.82 mA flows through each of the six resistors.

Step 4: Calculating Power Delivered by the Source

Our final step is to calculate the power delivered by the voltage source. Power (P) is the rate at which energy is transferred or used in a circuit. There are several formulas for calculating power, but the most convenient one for us is:

P = V * I

Where:

  • P = Power (in watts)
  • V = Voltage (in volts)
  • I = Current (in amperes)

We already know the voltage (120 V) and the current (0.00482 A). Plugging these values into the formula:

P = 120 V * 0.00482 A

P ≈ 0.5784 W

Therefore, the power delivered by the source is approximately 0.5784 watts. This means the voltage source is supplying energy to the circuit at a rate of about 0.5784 joules per second.

Step 5: Verifying Power Dissipation Across Resistors (Optional, but Recommended)

To further verify our results, we can calculate the power dissipated by each resistor and then sum these values. The total power dissipated by the resistors should ideally match the power delivered by the source (0.5784 W). This is due to the principle of conservation of energy – energy supplied by the source must equal the energy consumed by the circuit components.

The power dissipated by each resistor can be calculated using the formula:

P = I2 * R

Where:

  • P = Power (in watts)
  • I = Current (in amperes) – which is 0.00482 A for all resistors in our series circuit
  • R = Resistance (in ohms) – the individual resistance values for each resistor

Let's calculate the power dissipated by each resistor:

  • P1 = (0.00482 A)2 * 1250 Ω ≈ 0.0290 W
  • P2 = (0.00482 A)2 * 2120 Ω ≈ 0.0492 W
  • P3 = (0.00482 A)2 * 3330 Ω ≈ 0.0773 W
  • P4 = (0.00482 A)2 * 4650 Ω ≈ 0.1078 W
  • P5 = (0.00482 A)2 * 5330 Ω ≈ 0.1237 W
  • P6 = (0.00482 A)2 * 8330 Ω ≈ 0.1934 W

Now, let's add up the power dissipated by each resistor to find the total power dissipated:

PTotal Dissipated = P1 + P2 + P3 + P4 + P5 + P6

PTotal Dissipated ≈ 0.0290 W + 0.0492 W + 0.0773 W + 0.1078 W + 0.1237 W + 0.1934 W

PTotal Dissipated ≈ 0.5804 W

The total power dissipated by the resistors (0.5804 W) is very close to the power delivered by the source (0.5784 W). The slight difference is due to rounding errors in our calculations. This close agreement confirms that our calculations are accurate and reinforces the principle of conservation of energy.

Conclusion: Mastering Series Circuits

So, there you have it! We've successfully analyzed a series circuit, from drawing the diagram to calculating the power delivered by the source. We've used fundamental concepts like Ohm's Law and the power formula to understand how voltage, current, and resistance interact in a series circuit. Remember, in a series circuit, the current is the same throughout, the total resistance is the sum of individual resistances, and the power delivered by the source equals the total power dissipated by the resistors. By understanding these principles, you'll be well-equipped to tackle more complex circuit analysis problems in the future. Keep practicing, and you'll become a series circuit pro in no time!