Problem 62 - University Physics with Modern Physics

Question

A resistor with resistance \(R\) is connected to a battery that has emf 12.0 \(\mathrm{V}\) and internal resistance \(r=0.40 \Omega .\) For what two values of \(R\) will the power dissipated in the resistor be 80.0 \(\mathrm{W} ?\)

Step-by-Step Solution

Answer

Two possible resistance values will be calculated based on solving a quadratic equation derived from the power formula.

Step 1: Understand the question

The question asks: A resistor with resistance \(R\) is connected to a battery that has emf 12.0 \(\mathrm{V}\) and internal resistance \(r=0.40 \Omega .\) For what two values of \(R\) will the power dissipated in the resistor be 80.0 \(\mathrm{W} ?\)

Step 2: Apply Physics knowledge

Using fundamental Physics concepts to analyze the problem.

Step 3: Formulate the answer

Two possible resistance values will be calculated based on solving a quadratic equation derived from the power formula.

Step 4: Conclude

The answer is: Two possible resistance values will be calculated based on solving a quadratic equation derived from the power formula.

Key Concepts

Power Dissipation

Power dissipation is a key concept in electric circuits, indicating how much energy is converted into heat or work within a component. The power dissipated in a resistor is calculated using the formula: \[P = \frac{V^2}{R + r} \cdot R\]This formula involves three critical quantities: the voltage (\( V \)) provided by the battery, the resistance of the resistor (\( R \)), and the internal resistance of the battery (\( r \)). In this equation:

• \( P \) represents the power (in watts) dissipated.
• \( V \) is the electromotive force (emf) of the battery (given in volts).
• \( R \) is the resistance where power is dissipated.
• \( r \) is the internal resistance of the battery.

Understanding how power dissipation changes with varying resistance values is crucial for designing circuits that efficiently manage energy.

Resistance

Resistance is a measure of how much a component resists the flow of electric current. This property is quantified in ohms (\( \Omega \)). Every resistor has a specific resistance value, which determines how much current will flow through it for a given voltage.Key aspects of resistance to remember:

• High resistance means less current flow, lower power dissipation.
• Low resistance leads to increased current flow, higher power dissipation when voltage is constant.

The resistance value in a circuit affects the overall power consumption and energy efficiency. In our context, adjusting the resistance value \( R \) directly affects the power dissipation according to the formula provided, allowing for different levels of power management in circuits.

Internal Resistance

Internal resistance is an innate resistance found within all energy sources, like batteries. It's a property that affects how efficiently a battery can deliver power to a circuit.Here's how it impacts a circuit:

• A higher internal resistance reduces the effective voltage delivered to the load.
• It also alters how much power is dissipated in the external resistor \( R \).

In problems like the one given, internal resistance \( r \) plays a significant role in calculations. It must be accounted for to understand the real voltage and current values across a circuit, altering calculations for power dissipation and energy efficiency.

Ohm's Law

Ohm's Law is a fundamental principle in electronics, describing the relationship between voltage, current, and resistance. It is expressed as:\[V = I \cdot R\]In this equation:

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

Ohm's Law is crucial for understanding how different components in a circuit interact. It allows us to predict how changes in one quantity (like resistance) affect others (voltage, current). For example, if the resistance increases, the same voltage would result in less current flow, illustrating the balance between these electrical properties.