In the circuit shown in Fig. 𝐏 2 9 . 4 5, the capacitor has capacitance C=20 μF and is initially

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In the circuit shown in Fig. 𝐏 2 9 . 4 5, the capacitor has...

In the circuit shown in Fig. $\mathbf{P 2 9 . 4 5},$ the capacitor has capacitance $C=20 \mu \mathrm{F}$ and is initially charged to $100 \mathrm{~V}$ with the polarity shown. The resistor $R_{0}$ has resistance $10 \Omega$. At time $t=0$ the switch $S$ is closed. The small circuit is not connected in any way to the large one. The wire of the small circuit has a resistance of $1.0 \Omega / \mathrm{m}$ and contains 25 loops. The large circuit is a rectangle $2.0 \mathrm{~m}$ by $4.0 \mathrm{~m},$ while the small one has dimensions $a=10.0 \mathrm{~cm}$ and $b=20.0 \mathrm{~cm} .$ The distance $c$ is $5.0 \mathrm{~cm} .$ (The figure is not drawn to scale.) Both circuits are held stationary. Assume that only the wire nearest the small circuit produces an appreciable magnetic field through it. (a) Find the current in the large circuit $200 \mu$ s after $S$ is closed. (b) Find the current in the small circuit $200 \mu \mathrm{s}$ after $S$ is closed. (Hint: See Exercise $29.7 .$ ) (c) Find the direction of the current in the small circuit. (d) Justify why we can ignore the magnetic field from all the wires of the large circuit except for the wire closest to the small circuit.

Key Concepts

Magnetic Flux

Magnetic flux quantifies the total magnetic field passing through a given area and depends on both the magnitude of the magnetic field and the orientation of the area relative to the field. It is a critical quantity in electromagnetic induction, as its rate of change directly influences the magnitude of the induced EMF.

Approximations in Circuit Analysis

This concept involves recognizing when certain contributions, such as the magnetic effects from distant wires, can be neglected due to their minimal influence. By focusing on the dominant sources of magnetic field—typically those closest to the circuit of interest—one can simplify the analysis without sacrificing accuracy in predicting the circuit’s behavior.

Lenz's Law

Lenz's law provides the direction of any induced current resulting from a change in magnetic flux. It states that the induced current will flow in a direction such that its own magnetic field opposes the change in the original magnetic flux. This concept helps determine the direction of the induced current in a circuit experiencing a changing magnetic environment.

Magnetic Field from a Current-Carrying Conductor

This concept is centered on the magnetic field produced by electrical currents, particularly in conductors such as wires. Using principles like Ampere’s law or the Biot–Savart law, one can determine how the magnetic field varies with distance from a conductor. Understanding this decay is crucial when calculating how much of the magnetic field produced by one circuit influences another nearby circuit.

Faraday's Law of Electromagnetic Induction

Faraday's law states that a changing magnetic flux through a closed loop induces an electromotive force (EMF) in the loop, with the induced EMF equal to the negative rate of change of the magnetic flux. This principle is key to determining the transient induced currents in nearby circuits when the magnetic field from a discharging circuit changes over time.

RC Circuit Transients

This concept involves understanding how an RC (resistor-capacitor) circuit behaves when a capacitor discharges through a resistor. The voltage and current in the circuit decay exponentially with time, governed by a time constant ? = RC. This analysis is essential for predicting the circuit’s behavior at specific time intervals after a switching event.

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