Problem 54 - College Physics

Question

Lightbulbs in series, I. The power rating of a lightbulb is the power it consumes when connected across a 120 \(\mathrm{V}\) out-let. (a) If you put two 100 \(\mathrm{W}\) bulbs in series across a 120 \(\mathrm{V}\) outlet, how much power would each consume if its resistance were constant? (b) How much power does each one consume if you connect them in parallel across a 120 \(\mathrm{V}\) outlet?

Step-by-Step Solution

Answer

Each bulb consumes 20.83 W in series and 100 W in parallel.

Step 1: Understanding Bulb Resistance

First, we need to find the resistance of each 100 W bulb when connected to a 120 V outlet. Using the formula for power \(P = \frac{V^2}{R}\) where \(P\) is power, \(V\) is voltage, and \(R\) is resistance, we find \(R\) by rearranging to \(R = \frac{V^2}{P}\). Substituting the given values, \(R = \frac{120^2}{100} = 144\ \Omega\).

Step 2: Power Consumption in Series Connection

When two bulbs are in series, the total resistance \(R_{total}\) is the sum of the individual resistances, so \(R_{total} = 2 \times 144\ \Omega = 288\ \Omega\). The current \(I\) in the series circuit is given by \(I = \frac{V}{R_{total}} = \frac{120}{288} = \frac{5}{12}\ \text{A}\). The power consumed by each bulb is calculated using \(P = I^2 \times R = \left(\frac{5}{12}\right)^2 \times 144 = 20.83\ \text{W}\).

Step 3: Power Consumption in Parallel Connection

In a parallel connection, each bulb gets the full 120 V. The power consumed by each bulb remains \(P = \frac{V^2}{R} = \frac{120^2}{144} = 100\ \text{W}\) as they maintain their full power rating.

Key Concepts

Series Circuits

Series circuits are a common type of circuit configuration where electrical components are connected one after the other in a single path. When components like lightbulbs are arranged in series, the same current flows through each component. However, the total voltage across a series circuit is the sum of the voltages across each component.

- **Key Points about Series Circuits:** - The total resistance is the sum of individual resistances. - A single path for current means that if one component fails, the entire circuit stops working. - Adding more components increases total resistance, reducing the current in the circuit.
For example, if you have two lightbulbs of equal resistance in series, like in the exercise, the overall resistance increases. This reduces the current through the circuit, leading to lower power consumption for each bulb compared to when they are connected individually.

Parallel Circuits

Parallel circuits arrange components side by side, providing multiple paths for the current. Each component in a parallel circuit receives the full voltage of the power source, which allows them to operate independently.

- **Characteristics of Parallel Circuits:** - Total resistance decreases as you add more components because they provide additional paths for current. - Each component experiences the full voltage available, maintaining its individual functionality. - If one component fails, others continue working since they have their own pathways to the power source.
As seen in a practical example, such as the two lightbulbs in parallel, each bulb retains its full power rating since they each receive the full voltage of 120 V. This means both bulbs will consume their rated power of 100 W, functioning as though they are connected separately to the power source.

Resistance Calculation

Calculating resistance is crucial for understanding how circuits function. Resistance is a measure of how much a component resists the flow of current. It's calculated using the formula:

\[ R = \frac{V^2}{P} \]
Where: - \( R \) is resistance - \( V \) is voltage across the component - \( P \) is power consumed by the component
In our exercise, each 100 W lightbulb connected to a 120 V outlet has a resistance of 144 \( \Omega \). This was found by rearranging the power formula to solve for resistance, highlighting how voltage and power affect a device's resistance in an electric circuit.

Power Consumption

Power consumption refers to the rate at which electrical energy is converted into another form of energy, such as heat or light. It is calculated differently depending on how components are arranged in a circuit.

- **Power in Series Circuits:** - Power consumed by a component can be calculated using \( P = I^2 \times R \) where \( I \) is the current through the component and \( R \) is its resistance. - Since the current in a series circuit is reduced due to increased resistance, the power consumption of each individual component is typically less than its standalone rating.
- **Power in Parallel Circuits:** - Each component gets full voltage, so its power consumption remains at its rated value \( P = \frac{V^2}{R} \). - As seen in the example with two lightbulbs, each one consumes 100 W which is its full rated power.
Understanding these consumption patterns is vital for designing circuits that meet the desired energy requirements and performance specifications.