An experiment is conducted to measure the electrical...

An experiment is conducted to measure the electrical resistivity of Nichrome in the form of wires with different lengths and cross-sectional areas. For one set of measurements, a student uses 30-gauge wire, which has a cross-sectional area of $7.30 \times 10^{-8} \mathrm{m}^{2} .$ The student measures the potential difference across the wire and the current in the wire with a voltmeter and an ammeter, respectively. (a) For each set of measurements given in the table taken on wires of three different lengths, calculate the resistance of the wires and the corresponding values of the resistivity. (b) What is the average value of the resistivity? (c) Explain how this value compares with the value given in Table $27.2 .$ $$\begin{array}{lll} \hline {{L(\mathbf{m})}} & {\Delta V(\mathbf{V})} & {\mathbf{I}(\mathbf{A})} & {\mathbf{R}(\mathbf{\Omega})} & \rho(\Omega \cdot \mathrm{m})\\ \hline {0.540} & {5.22} & {0.72} \\ {1.028} & {5.82} & {0.414} \\ {1.543} & {5.94} & {0.281} \\ \hline\end{array}$$

An experiment is conducted to measure the electrical resistivity of Nichrome in the form of wires with different lengths and cross-sectional areas. For one set of measurements, a student uses 30-gauge wire, which has a cross-sectional area of $7.30 \times 10^{-8} \mathrm{m}^{2} .$ The student measures the potential difference across the wire and the current in the wire with a voltmeter and an ammeter, respectively. (a) For each set of measurements given in the table taken on wires of three different lengths, calculate the resistance of the wires and the corresponding values of the resistivity. (b) What is the average value of the resistivity? (c) Explain how this value compares with the value given in Table $27.2 .$
$$\begin{array}{lll} \hline {{L(\mathbf{m})}} & {\Delta V(\mathbf{V})} & {\mathbf{I}(\mathbf{A})} & {\mathbf{R}(\mathbf{\Omega})} & \rho(\Omega \cdot \mathrm{m})\\ \hline {0.540} & {5.22} & {0.72} \\ {1.028} & {5.82} & {0.414} \\ {1.543} & {5.94} & {0.281} \\ \hline\end{array}$$

Key Concepts

Experimental Measurement and Data Analysis

In the context of experimental physics, precise measurement of voltage, current, length, and cross-sectional area is crucial for accurately determining resistance and resistivity. Data analysis involves calculating these quantities, averaging multiple measurements, and comparing the results with standard or tabulated values to validate the experimental methods and the inherent properties of the material.

Ohm's Law

Ohm's law is a fundamental principle in electric circuit theory, stating that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance (V = IR). This law is critical for calculating resistance from experimental measurements of voltage and current.

Geometric Dependence of Resistance

The resistance of a conductor is not only determined by the material's resistivity but also by its geometry. The relationship R = ?·L/A illustrates that resistance increases with the length of the conductor and decreases with a larger cross-sectional area. This concept is key when designing experiments and analyzing how physical dimensions affect the measured resistance.

Electrical Resistivity

Electrical resistivity is an intrinsic property of a material that quantifies how strongly the material opposes the flow of electric current. It is defined independently of the sample's geometry and is calculated using the relation ? = R·A/L, where R is the resistance, A is the cross-sectional area, and L is the length. This concept helps in comparing different materials irrespective of their physical dimensions.

Electrical Resistance

Electrical resistance is a measure of the opposition to current flow through a conductor. It is influenced by both the material's resistivity and its physical dimensions, and is determined experimentally using Ohm’s law by measuring the voltage across and the current through the conductor. Understanding resistance is essential for analyzing circuits and material properties.

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