In the circuit shown in Fig. P29.47, the capacitor has capacitance C = 20 and is initially charg

In the circuit shown in Fig. P29.47, the capacitor has...

In the circuit shown in $\textbf{Fig. P29.47}$, the capacitor has capacitance $C =$ 20 $\mu$F and is initially charged to 100 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 $\Omega$/m and contains 25 loops. The large circuit is a rectangle 2.0 m by 4.0 m, while the small one has dimensions $a =$ 10.0 cm and $b =$ 20.0 cm. The distance c is 5.0 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 $\mu$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

Mutual Induction Between Circuits

Mutual induction discusses the phenomenon where a time-varying current in one circuit produces a changing magnetic field that induces an electromotive force in a neighboring circuit. This principle underlies the operation of devices like transformers and is fundamental in analyzing loosely coupled circuits, determining induced currents and understanding which parts of a system are significant in contributing to the magnetic flux through another circuit.

Lenz’s Law

Lenz’s law provides the direction of the induced current resulting from a change in magnetic flux. It states that the induced current will flow in such a way that its magnetic field opposes the change in the original magnetic flux. This concept is key in predicting the behavior of induced currents and is a direct consequence of the conservation of energy in electromagnetic processes.

Magnetic Field of a Current-Carrying Conductor

The magnetic field produced by a current-carrying wire can be determined through principles such as the Biot–Savart law or Ampère’s law. Understanding the distribution and strength of this magnetic field is essential when analyzing how one circuit can influence another via magnetic coupling, especially when only portions of a circuit contribute significantly to the overall magnetic flux.

Exponential Discharge in RC Circuits

When a capacitor in an RC circuit discharges, the current and voltage decay exponentially with time. The rate of this decay is determined by the circuit’s time constant (? = RC), which dictates how quickly the voltage drops and the current decreases. This principle is essential for predicting the behavior of the circuit at any given moment after a sudden change, such as when a switch is actuated.

RC Circuit Transients

This concept involves the study of how circuits containing resistors and capacitors respond over time when subject to sudden changes, such as closing a switch. The voltage across the capacitor and the current in the circuit change according to an exponential function, characterized by a time constant that is the product of the resistance and capacitance. This time-dependent behavior is crucial for understanding how energy stored in the capacitor dissipates through the resistor.

Faraday’s Law of Electromagnetic Induction

Faraday’s law explains how a change in magnetic flux through a circuit induces an electromotive force (EMF) in that circuit. This law is a cornerstone of electromagnetic theory and is applied to understand how a time-varying current in one circuit can produce a varying magnetic field, which in turn induces an EMF and a current in a nearby circuit.

Transcript

00:01 For this problem on the topic of electromagnetic induction, we are given a circuit as shown in the figure, and we are also given the capacitance and the voltage to which it is initially charged.

00:13 We have a resistor r0 to the given resistance, and we know at time t is equal to zero the switch is closed.

00:21 The small circuit in the diagram is not connected to the large one at all.

00:25 And we know for the small circuit the resistance per meter and the number of loops for the circuit.

00:34 Now the large circuit is a rectangle and the small one has given dimensions.

00:42 Now if both circuits are held stationary and only the wire nearest the small circuit produces an appreciable magnetic field, we are asked to calculate the current in the large circuit, the large circuit 200 microseconds after the switch is clear.

00:57 Closed, the current in the small circuit at the same time, the direction of the current in the small circuit, and then we are asked to justify why we can ignore the magnetic field from all the wires of the large circuit except for the closest one.

01:13 Now the current in a discharging rc circuit, i is equal to v0 over r, e to the minus t over r c, where v0 is the initial voltage across the capacitor.

01:25 Now for the small loop, the resistance is the number of loops 25 times 0 .6 meters times 1 oom per meter, which gives us 15 oms.

01:38 So let's begin to solve the problem.

01:41 So in part a, we know the large circuit is an rc circuit.

01:45 And an rc circuit has a time constant tor, which is equal to r times c.

01:53 And this is equal to a resistance, which is given as 10 oms times.

01:58 The capacitance of 20 microferrets, or 20 times 10 to the minus 6 ferrets, which gives us a time constant of 200 microseconds.

02:13 So the function, the current is as a function of time, i is equal to 100 volts divided by 10 oms times e to the minus t over 200 microseconds t over tall so if we let t be 200 microseconds we get the current i when t is equal to 200 microseconds to be 10 amperes times e to the minus 1 and so this current is 3 .7 amperes.

03:19 So that's the current in the large circuit at a time of 200 microseconds...

Please give Ace some feedback