/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 The following measurements of cu... [FREE SOLUTION] | 91Ó°ÊÓ

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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?

Short Answer

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

Step by step solution

01

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. \( I = 4.00 \), \( V = 15.52 \)
02

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.
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Treatment of Heart Failure. A heart defibrillator is used to enable the heart to start beating if it has stopped. This is done by passing a large current of 12 A through the body at 25 \(\mathrm{V}\) for a very short time, usually about 3.0 \(\mathrm{ms}\) . (a) What power does the defibrillator deliver to the body, and (b) how much energy is transferred?

Lightning Strikes. During lightning strikes from a cloud to the ground, currents as high as \(25,000\) A can occur and last for about 40\(\mu\) s. How much charge is transferred from the cloud to the earth during such a strike?

A ductile metal wire has resistance \(R .\) What will be the resistance of this wire in terms of \(R\) if it is stretched to three times its original length, assuming that the density and resistivity of the material do not change when the wire is stretched? (Hint: The amount of metal does not change, so stretching out the wire will affect its cross-sectional area.)

Copper has \(8.5 \times 10^{28}\) free electrons per cubic meter. A 71.0 -cm length of 12 -gauge copper wire that is 2.05 \(\mathrm{mm}\) in diameter carries 4.85 A of current. (a) How much time does it take for an electron to travel the length of the wire? (b) Repeat part (a) for 6-gauge copper wire (diameter 4.12 \(\mathrm{mm}\) ) of the same length that carries the same current.(c) Generally speaking, how does changing the diameter of a wire that carries a given amount of current affect the drift velocity of the electrons in the wire?

A Nichrome heating element that has resistance 28.0\(\Omega\) is connected to a battery that has emf 96.0 \(\mathrm{V}\) and internal resistance 1.2\(\Omega\) . An aluminum cup with mass 0.130 kg contains 0.200 \(\mathrm{kg}\) of water. The heating element is placed in the water and the electrical energy dissipated in the resistance of the heating element all goes into the cup and water. The element itself has very small mass. How much time does it take for the temperature of the cup and water to rise from \(21.2^{\circ} \mathrm{C}\) to \(34.5^{\circ} \mathrm{C}\) ? (The change of the resistance of the Nichrome due to its temperature change can be neglected.)

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