Problem 85 - College Physics

Question

Three identical resistors are connected in series. When a certain potential difference is applied across the combination, the total power dissipated is 27 \(\mathrm{W}\) . What power would be dissipated if the three resistors were connected in parallel across the same potential difference?

Step-by-Step Solution

Answer

The power dissipated in parallel is 243 W.

Step 1: Understand the Problem

We have three identical resistors connected in series, and we know that when a certain voltage is applied, the total power dissipated is 27 W. The question asks us to find the power dissipated when these resistors are connected in parallel with the same voltage applied.

Step 2: Recall the Formula for Power Dissipated

The power dissipated by a resistor can be calculated using the formula: \[ P = \frac{V^2}{R} \]where \( V \) is the voltage across the resistor and \( R \) is the resistance.

Step 3: Power for Series Connection

In series, the total resistance is the sum of individual resistances, i.e., \( R_s = 3R \). Thus, the power dissipated is:\[ P = \frac{V^2}{3R} = 27 \] This allows us to find \( R \) in terms of \( V \): \[ R = \frac{V^2}{81} \].

Step 4: Find Total Resistance for Parallel Connection

In parallel, the total resistance is given by: \[ \frac{1}{R_p} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{3}{R} \]So, \( R_p = \frac{R}{3} \).

Step 5: Calculate Power for Parallel Connection

Using the formula for power, the power dissipated in the parallel arrangement is:\[ P_p = \frac{V^2}{R_p} = \frac{V^2}{\frac{R}{3}} = 3 \cdot \frac{V^2}{R} \]We already know \( \frac{V^2}{R} = 81 \) (from Step 3), so:\[ P_p = 3 \times 81 = 243 \text{ W} \].

Step 6: Conclusion

When the three identical resistors are connected in parallel across the same potential difference, the power dissipated is 243 W.

Key Concepts

Series and parallel circuits

Imagine resistors as a chain of hurdles for electric current. They can be arranged in sequences called circuits. In a **series circuit**, all resistors are lined up one after another:

• Current has one pathway to flow.
• The total resistance increases as you add more resistors.

Think of it as water flowing through a pipe. If you make the pipe longer (adding more resistors), it becomes tougher for water to flow, thus more effort (or voltage) is needed. On the other hand, in a **parallel circuit**, resistors branch out:

• Current has multiple pathways to flow.
• The total resistance decreases as more resistors are added.

Imagine several pipes converging into one. Water (current) can select which route to take, facing less resistance overall.

Electric power calculation

To unravel electric power, think about the work done by electricity in a circuit. We determine it with the formula:

• \[ P = \frac{V^2}{R} \]

Where:

• \( P \) is the power in watts, reflecting energy consumption or dissipation.
• \( V \) represents voltage across the resistor.
• \( R \) is the resistance, opposing current flow.

These elements interplay just like pushing a box over a surface:

• **Voltage (\( V \))**: Like the push force on the box.
• **Resistance (\( R \))**: Like friction hindering movement.
• **Power (\( P \))**: The effort exerted to keep the box moving.

By controlling these factors, we balance the energy used and lost to keep devices running smoothly.

Resistor combinations

Resistors, often called the guardians of the circuit, can be combined in multiple ways to tailor circuit behavior. Understanding the impact of different combinations helps optimize electric circuits effectively.**Series Combination**:

• Resistors add up, resulting in: \( R_s = R_1 + R_2 + R_3 + ... \)
• The same current flows through each resistor, but the voltage varies across each one.
• Overall, uses more voltage to get the current moving.

**Parallel Combination**:

• Total resistance is less than any single resistor: \( \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... \)
• Voltage remains consistent across all resistors, but the current divides.
• Benefits in allowing more current to flow through the circuit.

These combinations impact how resistors handle electricity, determining energy usage and efficiency in circuits. Understanding them is like mastering a puzzle, ensuring all pieces fit precisely for optimal function.