In the circuit shown in Fig. 𝐏 29 . 4 5, the capacitor has capacitance C=20 μF and is initially

In the circuit shown in Fig. 𝐏 29 . 4 5, the capacitor has...

In the circuit shown in Fig. $\mathbf{P} 29 . \mathbf{4 5}$, 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 $200 \mu$ 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 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.

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.

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.

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.

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 Here i'm going to be examining fairday's law of induction in a specific example.

00:07 So i recall that fairday's law of induction says that the emf induced in a loop is equal to, and here i'll just find the magnitude of that, is equal to the number of turns in the loop times the change in magnetic flux inside the loop in time.

00:32 And again, we'll just think about the magnitude of that emf.

00:36 So this is an induced voltage that will tend to create a current in that detached loop.

00:44 And here the specific example we're going to look at is the wire is being driven by a time -bearing current.

00:52 And we're trying to determine the emf that's going to drive a current in the loop.

00:59 And here, recall that magnetic flux is equal to the product of magnetic field dotted into the area.

01:13 What's important to note is that b is created by the wire, and the area, of course, is the area of the loop.

01:32 So we're going to have to think about the magnetic field of the wire, which varies with distance from the wire.

01:38 Wire.

01:39 So if i use the right -hand rule, point my thumb downwards, i see that the magnetic field points into the page on one side and out on the side with a loop.

01:56 So the magnetic flux will be straight through that loop.

02:00 And furthermore, we can use a well -known ampere relationship for the magnitude of that magnetic field.

02:07 It is equal to mu -naut times i of aerole.

02:11 Of t in this case divided by 2 pi r.

02:17 So r is going with a distance from the loop, and that is going to be a variable.

02:29 So in order to work with the magnetic flux, we're going to have to realize that the magnetic field is a function of, in this case, we'll call it the x coordinate.

02:54 Times dx and the y direction.

02:58 We'll just call that the length of the loop on one side.

03:01 And that does not vary.

03:04 So we'll put in l in there for the d area.

03:11 And of course then we'll have to integrate over x to get the full magnetic flux from x1, which i'm showing as d1 in the picture to d2.

03:29 And we'll have to rewrite the magnetic field.

03:35 Okay, so there's a mu -not, i of t, does not care about the x direction, but the x is in the denominator.

03:54 So our full magnetic flux, in this case, is the length in the y direction that i show.

04:06 Mu -0, i of t, divided by 2 pi, and then we have the length.

04:12 The natural algorithm of d2 over d1.

04:20 And our emf is take the time derivative of that magnetic flux.

04:30 And of course, what is changing in time is the current.

04:35 And everything else, as long as the loop is not moving backwards and forwards.

04:42 That would be weird.

04:44 But i guess it could happen.

04:49 And then we have d .i.

04:51 By d t.

04:56 So remember that the wire must be driven by a time -bearing current in order for there to be an induced emf.

05:07 And we're assuming here that n is equal to one loop.

05:15 Okay, so now we're going to take a look at a particular example, maybe put some simple tricks in there.

05:24 So we're going to be driven by a capacitor of 20 micro -ferodays in a circuit with a resistor.

05:48 And there's a switch.

05:54 So there's 100 volts to begin with across the capacitor.

06:04 And so as we close that switch, what was going to happen is, let's put the polarity in there.

06:15 Our current will come down.

06:18 It will be a function of time.

06:21 And so we want to think about an rc circuit that the current is going to decay from its initial value.

06:32 And it's going to decay according to the rc time constant.

06:39 And so i -0 is 100 volts over 10 oms, and that is 10 amps.

06:56 D .i.

06:57 By d .t.

06:58 If we just take the derivative of that exponential is equal to i0, e to minus t over rc, and then minus 1 over rc.

07:10 So the current is slowing down.

07:18 And we'll just say that the current current decreases.

07:23 Okay, now we're going to put a very small loop, some distance over here to the side, and it has some little r, we'll call it, and it's got a side b and a side a and starts off 0 .05 meters away.

08:00 And we'll also say that there are 25 loops in this small loop.

08:12 And the resistance is per unit length is 1 oom per meter.

08:24 And what we would like to do is to find the induced...

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