Problem 40 - Fundamentals of Physics

Question

Thermal energy is produced in a resistor at a rate of \(100 \mathrm{~W}\) when the current is \(3.00 \mathrm{~A}\). What is the resistance?

Step-by-Step Solution

Answer

The resistance is approximately 11.11 Ω.

Step 1: Understand the Given Data

We are given the power produced by a resistor, which is 100 W, and the current flowing through it, which is 3.00 A. We need to find the resistance.

Step 2: Apply the Formula for Power

The formula for electrical power in terms of current and resistance is given by:\[ P = I^2 R \]where \( P \) is the power, \( I \) is the current, and \( R \) is the resistance. Here, \( P = 100 \,\text{W} \) and \( I = 3.00 \,\text{A} \).

Step 3: Rearrange the Formula to Solve for Resistance

We need to solve for \( R \). Rearrange the formula:\[ R = \frac{P}{I^2} \]

Step 4: Calculate the Resistance

Substitute the given values into the rearranged formula:\[ R = \frac{100}{3.00^2} \]Calculate \( 3.00^2 = 9 \) and then divide:\[ R = \frac{100}{9} \approx 11.11 \,\text{Ω} \]

Step 5: Result

The resistance of the resistor is approximately \( 11.11 \,\text{Ω} \).

Key Concepts

Electrical Resistance

Electric resistance is a crucial concept in understanding how circuits operate. It is a measure of how much a material or component in a circuit resists the flow of electric current. Think of it like the friction that a car experiences while moving on a road; similarly, resistance hinders current flow in a circuit. Any material with resistance will oppose the movement of electrons, which in turn generates heat.
Ohm's Law ties resistance directly to voltage and current through the formula: \( V = IR \), where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance.
In electronic devices, resistance is measured in ohms (symbol: Ω). The higher the resistance, the less current flows through the component for a given voltage. Components in circuits like resistors are used intentionally to control the amount of current. Identifying the resistance helps in designing circuits that effectively manage power consumption and prevent circuit damage.
It is important for students to remember that resistance is determined not only by the material of the component but also by its dimensions (length and cross-sectional area). A thicker and shorter wire, for instance, will have lower resistance compared to a thinner and longer one made from the same material.

Power Formula

The power formula is a cornerstone in electrical engineering and physics, helping to link the concepts of current, voltage, and resistance. Electric power, measured in watts (W), describes the rate at which energy is consumed or transformed in a circuit. In the exercise, the power formula is used in the form \( P = I^2 R \), where \( P \) is the power, \( I \) is the current, and \( R \) is the resistance.
This variant of the power formula is particularly useful when you know the current and resistance, but don't have the voltage value at hand. It shows that power is proportional to the square of the current multiplied by the resistance.
To break it down:

• If the current is increased, the power increases even more since it is squared in the formula.
• Higher resistance leads to higher power consumption for the same current.

An understanding of how to manipulate the power formula allows for efficient design and operation of electrical circuits, ensuring devices run smoothly without overheating or wasting energy. Knowing how to calculate power can help in selecting the appropriate components that align with the desired performance and efficiency, which is vital for both household electronics and industrial applications.

Current

Current is a fundamental concept in electricity that describes the flow of electric charge in a circuit. It is like the flow of water through a pipe, with electron flow constituting the current. Measured in amperes (A), current represents the amount of charge passing through a point in the circuit per unit of time.
Current can come in two forms: direct current (DC), which flows in one direction, and alternating current (AC), which periodically reverses direction. Regardless of the format, the amount of current determines the energy transferred in a circuit.
When considering Ohm's Law \( V = IR \), current is influenced by both the voltage applied and the resistance in the circuit. It is important to understand that:

• Higher voltage results in a higher current if resistance remains constant.
• Increasing resistance decreases the current if the voltage is fixed.

Understanding current flow helps in identifying potential issues in circuits, such as overloading, which can lead to overheating or damage to components. It allows for accurate calculations in circuit design, ensuring safety and efficiency in electrical systems from simple battery circuits to complex power grids.