On your first day at work as an electrical technician, you...

On your first day at work as an electrical technician, you are asked to determine the resistance per meter of a long piece of wire. The company you work for is poorly equipped. You find a battery, a voltmeter, and an ammeter, but no meter for directly measuring resistance (an ohmmeter). You put the leads from the voltmeter across the terminals of the battery, and the meter reads 12.6 V. You cut off a 20.0-m length of wire and connect it to the battery, with an ammeter in series with it to measure the current in the wire. The ammeter reads 7.00 A. You then cut off a 40.0-m length of wire and connect it to the battery, again with the ammeter in series to measure the current. The ammeter reads 4.20 A. Even though the equipment you have available to you is limited, your boss assures you of its high quality: The ammeter has very small resistance, and the voltmeter has very large resistance. What is the resistance of 1 meter of wire?

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical circuits that relates voltage, current, and resistance through the equation V = IR. This relationship allows one to calculate any one of these quantities if the other two are known, making it a central concept for analyzing and designing electrical circuits.

Internal Resistance

Internal resistance refers to the inherent resistance within a power source, such as a battery, which affects the actual voltage delivered to a circuit. Recognizing and accounting for a battery's internal resistance is crucial when calculating the true resistance of other components connected in series, as it contributes to the overall voltage drop in the circuit.

Series Circuit Analysis

Series circuit analysis involves understanding that when components are connected end-to-end, the same current flows through each element while the total voltage across the circuit is the sum of the individual voltage drops. This concept is key when determining how the resistance of individual components, including the battery’s internal resistance and other resistors, combine to affect the circuit’s behavior.

Parameter Identification Using Multiple Measurements

This concept involves taking multiple sets of measurements under different conditions to formulate a system of equations that can be solved for unknown parameters. By changing a variable in the circuit, such as the length of a wire (and thus its resistance), and measuring the resulting current, one can isolate and determine both the unknown resistor values and the internal resistance of the power source.

Transcript

00:02 Alright, so in this problem, so basically we have a battery and we have several pieces of wire and want to measure the resistance of the wire.

00:14 Okay, so the first measurement tells us that the voltage across the battery, v equal 12 .6 volts, okay? and the second measurement, so basically we put the battery connected with the wire.

00:36 So this is the circuit.

00:41 So the reason why we have two resistors, and the circuit is, so this one is the inner resistance of the battery, r.

00:49 And the other one, this one, this one, is the resistance of the wire, okay? and this is v.

00:54 So you can see that the current inside this loop is v2 -by -r plus r, right? and this current is given it's 7 ampires and the v is 12 .6 volts and in the second in the second measurement we have almost the same circuit but this is r primed and this is r this is v so the current i primed equal v diva by r plus r primed which is 4 .2 ampers okay so you can see that so we have the two equations listed.

01:37 And the relation between r and r prime is given here.

01:42 So because it is a wire, so wire has a relation of r equal r roll times l over s, right? so row is the resistivity and l is the lens of the wire, s is the cross section.

01:53 And r primed, in this case, equal row times l primed devout so come back to the two equations that we listed right here and we plug the relation of r and r prime into the two equations and we obtained the other two equations.

02:12 So v is 12 .6 divided by r plus so l in the first case is 20 meters, right? so 20 row divided by s equals 7 ampute...

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