Problem 18 - College Physics

Question

The following measurements of current and potential difference were made on a resistor constructed of Nichrome \(^{\mathrm{TM}}\) wire, where \(V_{a b}\) is the potential difference across the wire and \(I\) is the current through it: $$\begin{array}{ccccc}{I(\mathrm{A})} & {0.50} & {1.00} & {2.00} & {4.00} \\\ {V_{a b}(\mathrm{V})} & {1.94} & {3.88} & {7.76} & {15.52}\end{array}$$ (a) Graph \(V_{a b}\) as a function of \(I .\) (b) Does Ohm's law apply to Nichromet"y? How can you tell? (c) What is the resistance of the resistor in ohms?

Step-by-Step Solution

Answer

Ohm's law applies; resistance is 3.88 Ω.

Step 1: Plot Potential Difference as a Function of Current

To create the graph, plot each pair of values \( (I, V_{ab}) \):- For \( I = 0.50 \) A, \( V_{ab} = 1.94 \) V- For \( I = 1.00 \) A, \( V_{ab} = 3.88 \) V- For \( I = 2.00 \) A, \( V_{ab} = 7.76 \) V- For \( I = 4.00 \) A, \( V_{ab} = 15.52 \) VConnect these points with a straight line. The graph will depict \( V_{ab} \) (Y-axis) vs. \( I \) (X-axis), showing a linear relationship.

Step 2: Determine if Ohm's Law Applies

Ohm's law states that \( V = IR \), implying a linear relationship between voltage (\( V \)) and current (\( I \)). In the given data, we see that doubling the current doubles the voltage (eg. from \( 1.94 \) V to \( 3.88 \) V for current changing from \( 0.5 \) A to \( 1.0 \) A). The linearity of the graph confirms that Ohm’s law applies to the Nichrome wire in this case.

Step 3: Calculate the Resistance

The resistance \( R \) can be calculated using the formula \( R = \frac{V}{I} \). Using any data pair, say \( I = 1.00 \) A and \( V_{ab} = 3.88 \) V:\[ R = \frac{3.88}{1.00} = 3.88 \, \Omega \].The calculated resistance is consistent across all pairs due to linearity, confirming \( R = 3.88 \, \Omega \).

Key Concepts

Resistor

A resistor is a fundamental component in electrical circuits, designed to limit the flow of electric current. It provides a resistance to the passage of an electric current. This resistance is crucial for controlling the current within a circuit, thereby protecting devices, managing voltage, and conducting a host of other control functions. Resistors can be made of various materials, and they come in several types each suited to specific applications.

• Resistors are often color-coded to indicate their resistance value, tolerance, and sometimes, temperature coefficient.
• They are passive components, meaning they don't provide energy or gain in a circuit but absorb and release it.
• Resistors are commonly used to adjust signal levels, divide voltages, bias active elements, and terminate transmission lines.

When speaking about **Nichrome wire**, it acts as a resistor because it has a relatively high resistance compared with other conductors, which brings us to our next key topic.

Nichrome wire

Nichrome is an alloy made primarily of nickel and chromium, often used in applications that require consistent resistance, such as in heating elements and resistance wire. Its main feature is its ability to withstand high temperatures without oxidizing, thus making it ideal for applications like **electrical resistance heating** and resistors.

• Nichrome's stable resistance over a wide range of temperatures means it is less sensitive to temperature changes, thus maintaining linear performance.
• This linearity is crucial for precise applications where consistent performance is necessary.

When used in resistors, **Nichrome wire** maintains a predictable relationship between voltage and current, making it applicable in scenarios highlighting Ohm’s Law.

Electrical resistance

Electrical resistance is a measure of how much a material opposes the flow of electric current. The unit of measurement for resistance is the ohm (Ω), named after Georg Simon Ohm. He discovered that current is proportional to voltage across two points given a consistent resistance, described by **Ohm's Law**. The formula for Ohm's Law is: \[ V = I imes R \] where:

• **V** is the voltage across the resistor (in volts)
• **I** is the current through the resistor (in amperes)
• **R** is the resistance (in ohms)

The resistance of an object depends on its composition and temperature. Materials like Nichrome are engineered for stable resistance, as shown in the exercise where the resistance measured a constant \( 3.88 \, \Omega \), regardless of various currents applied. This consistency makes Nichrome an excellent material for applications that leverage Ohm’s Law.

Current-voltage graph

A current-voltage (IV) graph visually represents the relationship between the current flowing through a circuit and the voltage across it. This graph is critical for understanding how a resistor behaves under different electrical conditions.
When plotting an IV graph, current (I) is typically on the x-axis, while voltage (V) is on the y-axis. For a linear resistor, such as one made with Nichrome wire, the graph will show a straight line through the origin, indicating a direct proportionality between current and voltage.

• The slope of this line is the inverse of the resistance (1/R), and its consistency confirms the resistor’s adherence to Ohm’s Law.
• By analyzing the slope, we can determine the resistance from experimental data, which, in this exercise, was calculated as \( 3.88 \, \Omega \).

A linear IV graph not only verifies Ohm’s Law but also ensures predictable performance, which is paramount for designing efficient electronic circuits.