Problem 94 - Intermediate Algebra

Question

Electronics. The illustration below is a diagram of a resistor connected to a voltage source of 60 volts. As a result, the resistor loses power in the form of heat. The power \(P\) lost when a voltage \(E\) is placed across a resistance \(R\) (in ohms) is given by the formula $$ P=\frac{E^{2}}{R} $$ Solve for \(R\). If \(P\) is 4.8 watts and \(E\) is 60 volts, find \(R\). (IMAGE CANT COPY)

Step-by-Step Solution

Answer

The resistance \( R \) is 750 ohms.

Step 1: Understand the Formula

The formula for power lost as heat due to a resistor is given by \( P = \frac{E^2}{R} \). Here, \( P \) is the power in watts, \( E \) is the voltage in volts, and \( R \) is the resistance in ohms.

Step 2: Rearrange the Formula

We need to solve for \( R \). Start by multiplying both sides by \( R \) to get rid of the fraction: \( R \cdot P = E^2 \).Next, divide both sides by \( P \) to isolate \( R \): \( R = \frac{E^2}{P} \).

Step 3: Substitute Given Values

Substitute the given values into the equation. \( E = 60 \) volts and \( P = 4.8 \) watts.Substitute these into the equation: \( R = \frac{60^2}{4.8} \).

Step 4: Calculate the Resistance

First, calculate \( 60^2 \) which is 3600.Then, calculate \( \frac{3600}{4.8} \) to find \( R \).When you perform the division, \( R = 750 \) ohms.

Key Concepts

Resistance Calculation

In the world of electronics, knowing how to calculate resistance is a fundamental skill. Resistance is the property of a material that resists the flow of electric current, and it is measured in ohms (\( \Omega \)). In exercises like the one described, we need to calculate the resistance based on other known quantities, like voltage and power. To calculate resistance \( R \), simply use the rearranged formula given by the problem, which is \( R = \frac{E^2}{P} \). Here:

• \( E \) represents the voltage in volts.
• \( P \) indicates the power in watts.

Substitute the values for voltage and power in, and then carry out the calculations. This process demonstrates the direct relationship resistance has with these other electrical properties. In this exercise, plugging 60 volts and 4.8 watts into the equation gives \( R = 750 \Omega \), showing how resistance can be determined directly from the given conditions.

Power Formula

Electrical power, in simple terms, is the rate at which electrical work is done or energy is transferred. In our case, the power formula is expressed as \( P = \frac{E^2}{R} \). Breaking it down:

• \( P \) stands for power, measured in watts.
• \( E \) is the voltage across the resistor, in volts.
• \( R \) is the resistance of the resistor, in ohms.

This formula is a crucial concept in electronics as it helps in understanding how much power is dissipated. The exercise helps illustrate how to utilize this formula to solve for an unknown, showing how each component—voltage, resistance, and power—interacts within an electrical circuit. By rearranging it to solve for resistance, it highlights the inverse relationship between power loss and resistance when voltage remains constant.

Voltage Source

A voltage source is essential in an electric circuit as it provides the necessary potential difference needed to push current through the circuit. In the scenario provided in the exercise, the voltage source is 60 volts. Voltage can be thought of as the electrical "pressure" that causes electric charges to flow, with the resistor being the element that opposes this flow. Understanding how a voltage source interacts with other components, like a resistor, is key in circuit design and analysis. Higher voltage increases the potential amount of power transferred or dissipated in the circuit. Thus, in our exercise, when analyzing the effect of voltage set by a source, it's crucial to see how it impacts power and resistance within a given electrical setup.

Ohm's Law

Ohm's Law is a fundamental principle in electronics, stating that the current through a conductor between two points is directly proportional to the voltage across the two points. Symbolically, it can be expressed by the equation \( V = IR \), where:

• \( V \) is the voltage across the conductor in volts.
• \( I \) is the current through the conductor in amperes.
• \( R \) is the resistance of the conductor in ohms.

While Ohm's Law wasn't directly used in calculating the resistance in this exercise, understanding it helps in recognizing the relationships among voltage, current, and resistance. It supports the comprehension of how changing one quantity impacts the other two. As you learn about resistance calculations and power formulas more deeply, keeping Ohm's Law in mind will provide a broader context in understanding electrical circuits and their behavior.