Problem 37 - University Physics with Modern Physics

Question

The following measurements of current and potential difference were made on a resistor constructed of Nichrome wire: $$\begin{array}{llll}{\boldsymbol{I}(\mathbf{A})} & {0.50} & {1.00} & {2.00} & {4.00} \\ {\boldsymbol{V}_{a b}(\mathbf{V})} & {1.94} & {3.88} & {7.76} & {15.52}\end{array}$$ (a) Graph \(V_{a b}\) as a function of \(I\) (b) Does Nichrome obey Ohm's law? How can you tell? (c) What is the resistance of the resistor in ohms?

Step-by-Step Solution

Answer

Nichrome obeys Ohm's law with a resistance of 3.88 ohms.

Step 1: Organize the Data

List out the given data points in terms of current (I) in amperes and voltage (V) in volts:1. \( I = 0.50 \), \( V = 1.94 \)2. \( I = 1.00 \), \( V = 3.88 \)3. \( I = 2.00 \), \( V = 7.76 \)4. \( I = 4.00 \), \( V = 15.52 \)

Step 2: Create a Graph

Plot the voltage data \(V_{ab}\) on the y-axis and the current data \(I\) on the x-axis. Create a graph with these points:- (0.50, 1.94)- (1.00, 3.88)- (2.00, 7.76)- (4.00, 15.52). Observe the trend of the line. It should be a straight line passing through the origin.

Step 3: Analyze the Graph for Ohm's Law

Ohm's Law states that \( V = IR \), where a linear relationship through the origin indicates a constant resistance (R). As the plot is a straight line through the origin, Nichrome obeys Ohm's Law.

Step 4: Calculate the Resistance

Calculate the resistance using the formula \( R = \frac{V}{I} \). Use any data point for consistency:For \( I = 0.50 \) and \( V = 1.94 \), \( R = \frac{1.94}{0.50} = 3.88 \, \text{ohms} \).Repeat calculation with another data point, for example \( I = 1.00 \) and \( V = 3.88 \),\( R = \frac{3.88}{1.00} = 3.88 \, \text{ohms} \). The resistance remains constant.

Key Concepts

Nichrome Wire

Nichrome is a popular material used in various practical applications due to its particular electrical properties. It is an alloy composed mainly of nickel and chromium, which gives it high resistance and a stable performance even at high temperatures.
This quality makes Nichrome an excellent choice for applications such as heating elements and resistors. When electric current passes through a Nichrome wire, it offers more resistance than many other materials, generating heat efficiently.

• High resistance: This characteristic makes Nichrome suitable for devices where heat generation is essential.
• Temperature stability: Nichrome maintains its structural integrity and electrical resistance over a wide temperature range.
• Durability: It is resistant to oxidation and corrosion, extending the lifespan of Nichrome-based components.

Nichrome’s properties are why it's used in industries ranging from electronics to aeronautics. Understanding its behavior in electrical circuits, such as its adherence to Ohm's Law, can be critical for designing reliable and efficient electrical systems.

Electrical Resistance

Electrical resistance is a fundamental concept in understanding how electricity flows through materials. It is defined as the opposition a material offers to the flow of electric current. The higher the resistance, the less current will flow through the material at a given voltage.
The unit of electrical resistance is the ohm (Ω). To calculate resistance, Ohm's Law is employed, which is expressed as: \[ R = \frac{V}{I} \] Where \( R \) is resistance, \( V \) is voltage, and \( I \) is current.

• Materials like Nichrome are chosen for specific tasks because of their precise resistance values.
• In practical applications, controlling resistance can help manage energy consumption and heat generation.
• Resistance plays a critical role in designing circuits and selecting materials.

By comprehending the resistance of different materials, such as Nichrome wire, engineers can tailor circuits to achieve desired electrical performance. This understanding also aids in analyzing the efficiency and safety of electrical devices.

Current-Voltage Relationship

The current-voltage relationship is a fundamental principle in electrical circuits, dictating how voltage across a material affects the current flowing through it. Ohm’s Law is the guiding formula for understanding this relationship in many conductive materials.
Ohm's Law states that the current \( I \), through a conductor between two points, is directly proportional to the voltage \( V \) across the two points, which can be expressed as: \[ V = IR \] Where \( V \) is voltage, \( I \) is current, and \( R \) is resistance. This relationship implies that, for a constant resistance, a doubling in voltage will result in a doubling of current.

• A plot of voltage against current results in a straight line for materials obeying Ohm’s Law.
• A constant slope on this graph indicates consistent resistance, as observed with Nichrome wire.
• Understanding this relationship helps in predicting how circuits will respond under different conditions.

This linear relationship affirms the predictability and reliability of the material's electrical behavior, crucial for designing and analyzing circuits effectively.