A sequence of potential differences V is applied across a...

A sequence of potential differences $V$ is applied across a wire and the (diameter = 0.32mm, length = 11 cm) resulting currents $I$ are measured as follows: ($a$) If this wire obeys Ohm's law, graphing $I$ vs. V will result in a straight-line plot. Explain why this is so and determine the theoretical predictions for the straight line's slope and $y$-intercept. ($b$) Plot $I$ vs. $V$. Based on this plot, can you conclude that the wire obeys Ohm's law (i.e., did you obtain a straight line with the expected $y$-intercept, within the values of the significant figures)? If so, determine the wire's resistance $R$. ($c$) Calculate the wire's resistivity and use Table 18-1 to identify the solid material from which it is composed.

A sequence of potential differences $V$ is applied across a wire and the (diameter = 0.32mm, length = 11 cm) resulting currents $I$ are measured as follows:

($a$) If this wire obeys Ohm's law, graphing $I$ vs. V will result in a straight-line plot. Explain why this is so and determine the theoretical predictions for the straight line's slope and $y$-intercept. ($b$) Plot $I$ vs. $V$. Based on this plot, can you conclude that the wire obeys Ohm's law (i.e., did you obtain a straight line with the expected $y$-intercept, within the values of the significant figures)? If so, determine the wire's resistance $R$. ($c$) Calculate the wire's resistivity and use Table 18-1 to identify the solid material from which it is composed.

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical circuits which states that the current through a conductor between two points is directly proportional to the voltage across the two points, provided the temperature and other physical conditions remain constant. This linear relationship, defined by V = IR, means that if a material obeys Ohm's Law, a plot of current versus voltage should yield a straight line, with the slope related to the inverse of the resistance.

Resistance

Resistance is a property of a material or component that quantifies its opposition to the flow of electric current. It is calculated using the relationship R = V/I, where V is the voltage across the component and I is the current flowing through it. In a system obeying Ohm's Law, the resistance remains constant regardless of the applied voltage, leading to a linear current versus voltage graph where the slope is 1/R.

Resistivity

Resistivity is an intrinsic property of a material that indicates how strongly it resists current flow. It is related to the resistance R of a wire by the formula R = ?L/A, where L is the length of the wire, A is its cross-sectional area, and ? is the resistivity. By measuring the electrical resistance and knowing the dimensions of the wire, one can calculate the material’s resistivity, which is useful for material identification.

Graphical Analysis of I-V Data

Graphical analysis involves plotting the measured current against the applied voltage to determine if the relationship is linear, which is expected for a resistor obeying Ohm's Law. The slope of the resulting line indicates the inverse of the resistance and the y-intercept should ideally be zero or near zero. This method is essential for verifying the validity of Ohm's Law and for determining the resistance of the component experimentally.

Measurement Accuracy and Significant Figures

Examining the y-intercept and comparing measured data with theoretical predictions involves careful attention to measurement accuracy and significant figures. In experimental physics, the degree of uncertainty or error in the measurements is important for determining whether observed deviations from the expected behavior (such as a non-zero y-intercept) are statistically significant or within the margins of experimental error.

Transcript

00:01 So here we have a wire.

00:01 We have a diameter of 0 .32 millimeters.

00:06 We can say that this is 3 .2 times 10 to negative 4 meters.

00:12 And we have a length of 11 centimeters or 0 .11 meters.

00:19 So we also have a table.

00:22 So we have vv and then i in millie amps all the way to 0 .5.

00:36 I apologize.

00:42 And then here it would be 72, 142, 218, 290, and 357.

00:56 So at this point, we can say, if this follows alms law, where the voltage is equal to the product of the current and the resistance, we can say that we can compare this, solve for i, this is going to equal v over r, say that i equals our y input essentially and our v is our x so i would be our um dependent variable and our x would be our independent variable um so we would essentially compare this to y equals mx plus b and m here the slope is going to be one over r and of course b equals zero so b will be the y intercept so the y intercept will essentially equal zero so um this would be essentially this would be the slope if the slope and the y intercept if this wire um obeys oms law so at this point we should graph it and we'll have um voltage and volts rather i apologize this will be current in milliams this will be our y and then our x our x axis will be volts and uh the voltage in volts so we have one two three four so here it would be point two point four point six and then here it'll be 100 200 300 300 400 so uh we it's essentially going to be point one at 72 so it's 42, so around here.

03:01 Point 3 at 218, 0 .4 at 290, and then 0 .5 at 357.

03:12 So essentially it'll look like this with our points.

03:23 And as you can see here, you can use excel or a t -i -84 or t -85 in order to find the slope of this.

03:32 So here the slope m is going to be equal.

03:36 To 718 millie amps per volt so i would simply be equal to 718 millie amps per volts times v the voltage again this follows um's law because again b equals zero so the y intercept is going to be equal to zero and then we can say to find the resistance of the wire we can say that r is going to be equal to 1 .718 amps per volt...

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