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Chapter 7: Problem 6

A rectangular loop of wire is situated so that one end (height \(h\) ) is between the plates of a paralle]-plate capacitor (Fig. 7.9), oriented parallel to the field \(\mathbf{E}\). The other cnd is way outside, where the field is essentially zero. What is the emf in this loop? If the total resistance is \(R\), what current flows? Explain. [Warming: this is a trick question. so be careful: if you have invented a perpetual motion machine, there's probably something wrong with it.]

Short Answer

Expert verified
The emf in the loop is zero, therefore, no current flows through it.

Step by step solution

01

Understanding the System

We have a rectangular loop of wire partially inside a capacitor with a uniform electric field \( \mathbf{E} \). The upper part of the loop is outside of the capacitor where the electric field is zero. We need to find the electromotive force (emf) in the loop and the current given the loop's total resistance \( R \).
02

Analyzing the Electric Field Effect

The portion of the wire within the capacitor experiences the electric field \( \mathbf{E} \). The emf is due to the work done by this electric field on charges as they move around the loop. But because the wire completes a circuit both inside and outside the field, this effect cancels out.
03

Calculating the EMF

The emf in the loop can be calculated as \( \text{emf} = \int \mathbf{E} \cdot d\mathbf{l} \). However, because the wire moves through equal and opposing fields that cancel, and wires in a static field do not induce emf in ideal situations, the net emf is effectively zero.
04

Determining the Current

Since the emf in the loop is zero, according to Ohm's Law \( I = \frac{\text{emf}}{R} \), the current \( I \) is zero as well. The absence of a net emf means no current flows through the loop.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Field
An electric field is a force field surrounding a charged particle. It is defined as the electric force per unit charge that would be experienced by a stationary point charge at any given point in space. In equations, it is expressed as \( \mathbf{E} = \frac{\mathbf{F}}{q} \), where \( \mathbf{F} \) is the force exerted by the field, and \( q \) is the charge experiencing the field.
When you place a charged particle in an electric field, it will experience a force. The direction of this force depends on the sign of the charge and the orientation of the field lines.
In a uniform electric field, such as the one in a capacitor, the field lines are parallel and equally spaced. This results in a constant magnitude and direction of the electric field within the area between the capacitor plates.
  • The electric field inside this area does the work on the charges, influencing potential energy and charge motion within conductors like our wire loop.
Ohm's Law
Ohm's Law is a fundamental principle of electricity that relates the current flowing through a conductor to the voltage across it, and the resistance it encounters. This law is expressed through the equation \( V = IR \), where \( V \) stands for voltage (or potential difference), \( I \) is the current, and \( R \) is the resistance.
This relationship implies that the current flowing through a circuit is directly proportional to the voltage applied across it and inversely proportional to its resistance.
In our wire loop scenario, Ohm's Law helps us understand why no current flows when the emf is zero. Without a driving voltage, as determined by the emf in the loop, Ohm's Law tells us that the current must also be zero.
  • As such, the current can only flow if there is a non-zero emf driving it, which is not the case in this instance.
Resistance
Resistance is a measure of the opposition to current flow in an electrical circuit. It limits how much current can flow through a material given a particular voltage. Resistance is measured in ohms (\( \Omega \)), named after the physicist Georg Simon Ohm.
Every material presents some level of resistance, whether it's a highly resistive insulator or a moderately resistive conductor. The total resistance in a circuit affects how much current flows for a given voltage, as per Ohm's Law.
  • Materials with high resistance require more energy to push current through, while low-resistance materials allow easier current flow.
  • In the given loop problem, despite the presence of resistance \( R \), no current flows due to a zero emf.
Current Flow
Current flow, succinctly, is the movement of electric charge carriers (usually electrons) through a conductor like wire. Current is measured in amperes (A), with direction defined from positive to negative, although electrons flow opposite this direction.
The flow of current requires an electromotive force (emf), or voltage, to propel the charge carriers. Without a driving force, as in the scenario where emf is zero, charges remain static, and no current flows.
  • Understanding this principle helps demonstrate why our wire loop, even with resistance, carries no current without an emf.
  • In many practical circuits, maintaining steady current flow involves ensuring both adequate voltage and minimal resistance to efficiently power devices.

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Most popular questions from this chapter

\(\mathrm{~A}\) square loop of wire, of side \(a\), lies midway between two long wires, \(3 a\) apart. and in the same planc. (Actually, the long wires are sides of a large rectangular loop. but the short ends are so far away that they can be neglected.) A clockwise current \(I\) in the square loop is gradually increasing: \(d I / d t=k\) (a constant). Find the emf induced in the big loop. Which way will the induced current flow?

A square loop (side \(a\) ) is mounted on a vertical shaft and rotated at angular velocity \(\omega\) (Fig. 7.18). A uniform magnetic field \(\mathbf{B}\) points to the right. Find the \(\mathcal{E}(t)\) for this alternating current generator.

An alternating current \(I_{0} \cos (\omega t)\) (amplitude \(0.5 \mathrm{~A}\), frequency \(60 \mathrm{~Hz}\) ) flows down a straight wire, which runs along the axis of a toroidal coil with rectangular cross section (inner radius \(1 \mathrm{~cm}\), outer radius \(2 \mathrm{~cm}\), height \(1 \mathrm{~cm}, 1000\) tums). The coil is connected to a 500 \(\Omega\) resistor. (a) In the quasistatic approximation, what emf is induced in the toroid? Find the current, \(I_{r}(t)\). in the resistor. (b) Calculate the back emf in the coil, due to the current \(I_{r}(t) .\) What is the ratio of the amplitudes of this back emf and the "direct" emf in (a)?

(a) Two metal objects are embedded in weakly conducting material of conductivity \(\sigma\) (Fig. 7.6). Show that the resistance between them is related to the capacitance of the arrangement by $$R=\frac{\epsilon_{0}}{\sigma C}$$ (b) Suppose you connected a battery between 1 and 2 and charged them up to a potential difference \(V_{0}\). If you then disconnect the battery, the charge will gradually leak off. Show that \(V(t)=V_{0} e^{-f / \mathrm{r}}\), and find the time constant, \(r\), in terms of \(\epsilon_{0}\) and \(\sigma\).

Problem 7.59 Prove Alfven's theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, "frozen" into the fluid.) (a) Use Ohm's law, in the form of Eq. 7.2, together with Faraday's law, to prove that if \(\sigma=\infty\) and \(\mathbf{J}\) is finite, then $$\frac{\partial \mathbf{B}}{\partial t}=\nabla \times(\mathbf{v} \times \mathbf{B})$$ (b) Let \(\mathcal{S}\) be the surface bounded by the loop \((\mathcal{P})\) at time \(t\), and \(\mathcal{S}^{\prime}\) a surface bounded by the loop in its new position \(\left(\mathcal{P}^{\prime}\right)\) at time \(t+d t\) (sec Fig. 7.56). The change in flux is $$d \Phi=\int_{\mathcal{S}^{\prime}} \mathbf{B}(t+d t) \cdot d \mathbf{a}-\int_{\mathcal{S}} \mathbf{B}(t) \cdot d \mathbf{a} .$$
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