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Chapter 25: Problem 40

The following measurements were made on a Thyrite resistor: $$ \begin{array}{l|llll}{I(A)} & {0.50} & {1.00} & {2.00} & {4.00} \\\ {V_{\omega}(V)} & {2.55} & {3.11} & {3.77} & {4.58}\end{array} $$ (a) Graph \(V_{a b}\) as a function of \(I .\) (b) Does Thyrite obey Ohm's law? How can you tell?(c) Graph the resistance \(R=V_{a b} / I\) as a function of \(I .\)

Short Answer

Expert verified
Thyrite does not obey Ohm's Law; the relationship is non-linear, and resistance changes with current.

Step by step solution

01

Understand the relationship between variables

The data provides the current \( I \) in amperes and the voltage \( V_{ab} \) in volts across a Thyrite resistor. We need to graph and analyze how the voltage depends on the current and calculate resistance based on Ohm's Law.
02

Graph Voltage vs. Current

To graph \( V_{ab} \) as a function of \( I \), plot the given current values \((0.50, 1.00, 2.00, 4.00)\) on the x-axis and the corresponding voltage values \((2.55, 3.11, 3.77, 4.58)\) on the y-axis. Connect these points to visualize the relationship.
03

Assess Ohm's Law

Ohm's Law states that \( V = IR \), which implies a linear relationship between voltage and current (a straight line graph through the origin if resistance is constant). Examine the plotted points from Step 2 to see if they form a straight line through the origin.
04

Graph Resistance vs. Current

Calculate the resistance for each pair of current and voltage using the formula \( R = \frac{V_{ab}}{I} \).- For \( I = 0.50 \, A \), \( R = \frac{2.55}{0.50} = 5.10 \, \Omega \)- For \( I = 1.00 \, A \), \( R = \frac{3.11}{1.00} = 3.11 \, \Omega \)- For \( I = 2.00 \, A \), \( R = \frac{3.77}{2.00} = 1.885 \, \Omega \)- For \( I = 4.00 \, A \), \( R = \frac{4.58}{4.00} = 1.145 \, \Omega \)Plot these resistance values against the current on a graph.
05

Analyze and Conclude

The resistance graph is not constant, and the voltage vs. current graph shows a curve rather than a straight line through the origin. This indicates that the Thyrite resistor does not obey Ohm's Law, which states that resistance should remain constant for a linear relationship between voltage and current.

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Key Concepts

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

Voltage-Current Relationship
Understanding the relationship between voltage and current is at the core of analyzing electrical circuits. Ohm's Law serves as the basic principle here, defined as \( V = IR \). This equation suggests that, for a given resistor, the voltage across it (\( V \)) should increase linearly with the current (\( I \)) passing through it. However, this relationship holds true primarily for ohmic materials, where the resistance \( R \) remains constant.
By plotting voltage as a function of current, we can visualize this relationship. Ideally, if a material obeys Ohm's Law, plotting these points results in a straight line through the origin, indicating that the voltage increases proportionally with the current. In the case of the Thyrite resistor mentioned in the exercise, the relationship between voltage and current is non-linear. The points on the graph do not form a straight line, suggesting that the resistor behaves differently than an ohmic resistor.
This deviation from linearity is crucial because it signifies how certain materials respond under varying electrical conditions, which affects how they are deployed in circuits.
Resistance Calculation
Resistance is a measure of how much an object opposes the flow of electric current. According to Ohm's Law, resistance can be calculated using the formula \( R = \frac{V}{I} \). This formula provides insight into whether a material maintains constant resistance as the current changes, which is a defining property of ohmic materials.
In the provided exercise, the resistance of the Thyrite resistor is calculated at various points by taking the given voltage and dividing it by the corresponding current. For example:
  • At \( I = 0.50 \) A, the resistance \( R = 5.10 \) 惟
  • At \( I = 1.00 \) A, \( R = 3.11 \) 惟
  • At \( I = 2.00 \) A, \( R = 1.885 \) 惟
  • At \( I = 4.00 \) A, \( R = 1.145 \) 惟
These calculations reveal a decrease in resistance with increasing current, indicating variable resistance. This change in resistance is typical of non-ohmic materials, where the resistance does not remain constant.
Non-Ohmic Resistor Behavior
Non-ohmic resistor behavior is characterized by the deviation from the constant resistance expected from Ohm's Law. Materials exhibiting this behavior do not show a linear relationship between voltage and current, meaning their resistance varies with changes in current.
The Thyrite resistor, used as an example in the exercise, falls into this category. As the data suggests, plotting the resistance against current reveals a downward trend in resistance with increasing current.
This behavior can be due to several factors, such as temperature changes, material composition, or intrinsic properties of the component. Understanding non-ohmic behavior is crucial in specific applications where predictable resistance across varying conditions is not feasible. Such materials are widely used in situations where their resistance should adapt dynamically to the current, like in surge protectors or other protective devices.
In summary, recognizing non-ohmic resistor behavior helps engineers and students predict how a system will behave under different electrical currents and design circuits that accommodate these changes effectively.

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Most popular questions from this chapter

The Tolman-Stewart experiment in 1916 demonstrated that the free charges in a metal have negative charge and provided a quantitative measurement of their charge-to-mass ratio, \(|q| / m\) . The experiment consisted of abruptly stopping a rapidly rotating spool of wire and measuring the potential difference that this produced between the ends of the wire. In a simplified model of this experment, consider a metal rod of length \(L\) that is given a uniform acceleration \(\overrightarrow{\boldsymbol{d}}\) to the right. Initially the free charges in the metal lag behind the rod's motion, thus setting up an electric field \(\overrightarrow{\boldsymbol{E}}\) in the rod. In the steady state this field exerts a force on the free charges that makes them accelerate along with the rod. (a) Apply \(\Sigma \vec{F}=m \vec{d}\) to the free charges to obtain an expression for \(|q| / m\) in terms of the magnitudes of the induced electric field \(\overrightarrow{\boldsymbol{k}}\) and the acceleration \(\overrightarrow{\boldsymbol{d}} .\) (b) If all the free charges in the metal rod have the same acceleration, the electric field \(\overrightarrow{\boldsymbol{E}}\) is the same at all points in the rod. Use this fact to rewrite the expression for \(|q| / m\) in terms of the potential \(V_{b c}\) between the ends of the rod (Fig. 25.44\()\) . (c) If the free charges have negative charge, which end of the rod, \(b\) or \(c,\) is at higher potential? (d) If the rod is 0.50 \(\mathrm{m}\) long and the free charges are electrons (charge \(q=-1.60 \times 10^{-19} \mathrm{C},\) mass \(9.11 \times 10^{-31} \mathrm{kg} ),\) what magnitude of acceleration is required to produce a potential difference of 1.0 \(\mathrm{mV}\) between the ends of the rod?(e) Discuss why the actual experiment used a rotating spool of thin wire rather than a moving bar as in our simplified analysis.

The current in a wire varies with time according to the relationship \(I=55 \mathrm{A}-\left(0.65 \mathrm{A} / \mathrm{s}^{2}\right) t^{2}\) . (a) How many coulombs of charge pass a cross section of the wire in the time interval between \(t=0\) and \(t=8.0 \mathrm{s} ?(\mathrm{b})\) What constant current would transport the same charge in the same time interval?

Copper has \(8.5 \times 10^{28}\) free electrons per cubic meter. A 71.0 - \(\mathrm{cm}\) length of 12 -gauge copper wire that is 2.05 \(\mathrm{mm}\) in diameter carries 4.85 \(\mathrm{A}\) 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 \(1.50-m\) cylindrical rod of diameter 0.500 \(\mathrm{cm}\) is connected to a power supply that maintains a constant potential difference of 15.0 \(\mathrm{V}\) across its ends, while an ammeter measures the current through it. You observe that at room temperature \(\left(20.0^{\circ} \mathrm{C}\right)\) the ammeter reads 18.5 \(\mathrm{A}\) , while at \(92.0^{\circ} \mathrm{C}\) it reads 17.2 \(\mathrm{A}\) . You can ignore any thermal expansion of the rod. Find (a) the resistivity and (b) the temperature coefficient of resistivity at \(20^{\circ} \mathrm{C}\) for the material of the rod.

You apply a potential difference of 4.50 \(\mathrm{V}\) between the ends of a wire that is 2.50 \(\mathrm{m}\) in length and 0.654 \(\mathrm{mm}\) in radius. The resulting current through the wire is 17.6 \(\mathrm{A}\) . What is the resistivity of the wire?
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