You have a 100-m copper wire with cross-sectional area 0.20 mm^2 and resistivity 1.7 ×10^-8 Ω·m

step 1 Hi, in the given problem we are having a copper wire whose length is given as 100 meter and its

You have a 100-m copper wire with cross-sectional area 0.20...

You have a $100-\mathrm{m}$ copper wire with cross-sectional area $0.20 \mathrm{mm}^{2}$ and resistivity $1.7 \times 10^{-8} \Omega \cdot \mathrm{m} .$ You make a 780 -turn solenoid by winding the wire on a plastic tube (diameter $4.0 \mathrm{cm},$ length $10.0 \mathrm{cm}$ ). Determine (a) the resistance of the solenoid, (b) the inductance of the solenoid, and (c) the frequency of the alternating current at which the inductive reactance of the solenoid is equal to its resistance.

You have a $100-\mathrm{m}$ copper wire with cross-sectional area $0.20 \mathrm{mm}^{2}$ and resistivity $1.7 \times 10^{-8} \Omega \cdot \mathrm{m} .$ You make a 780 -turn solenoid by winding the wire on a plastic tube (diameter $4.0 \mathrm{cm},$ length $10.0 \mathrm{cm}$ ). Determine
(a) the resistance of the solenoid, (b) the inductance of the solenoid, and
(c) the frequency of the alternating current at which the inductive reactance of the solenoid is equal to its resistance.

Key Concepts

Resistivity and Resistance Calculation

Resistance is an intrinsic property of a material determined by its resistivity, the length of the conductor, and its cross?sectional area. In practice, the resistance of a wire is calculated using the formula R = ?L/A, where ? is the resistivity, L is the length, and A is the cross-sectional area. This concept is fundamental to understanding how material properties and geometry influence the electric conduction in circuits.

Inductance of a Solenoid

Inductance quantifies the ability of a solenoid or coil to store energy in its magnetic field when current flows through it. For a solenoid, the inductance is typically calculated using the formula L = ??N²A/?, where ?? is the permeability of free space, N is the number of turns, A is the cross-sectional area of the solenoid, and ? is its length. This key concept is central in electromagnetism, especially in designing circuits that involve magnetic energy storage and filtering.

Inductive Reactance in AC Circuits

Inductive reactance is the opposition offered by an inductor to alternating current and is frequency-dependent, given by X_L = ?L or 2?fL, where ? is the angular frequency and L is the inductance. It plays a critical role in AC circuit analysis by determining how an inductor impedes current flow at different frequencies, influencing the circuit’s phase and amplitude responses.

AC Circuit Analysis and Frequency Matching

In AC circuit analysis, one often needs to find the frequency at which the inductive reactance equals the resistance, as this condition affects the phase relationship between voltage and current. Equating X_L (2?fL) to the resistance R allows determination of the specific frequency at which these two opposing effects balance, which is important in applications like tuning circuits and impedance matching.

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