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Chapter 29: Problem 15

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in Fig. E29.15. The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) \(B\) is increasing; (b) \(B\) is decreasing; (c) \(B\) is constant with value \(B_{0} .\) Explain your reasoning.

Short Answer

Expert verified
a) Counterclockwise, b) Clockwise, c) No current.

Step by step solution

01

Understand Faraday's Law

Faraday's Law of electromagnetic induction states that a change in magnetic flux through a circuit induces an electromotive force (emf) in the circuit. This can lead to an induced current in a closed conducting loop. Mathematically, the induced emf is given by \( \mathcal{E} = -\frac{d\Phi_B}{dt} \), where \( \Phi_B \) is the magnetic flux.
02

Apply Lenz's Law

Lenz's Law tells us that the direction of the induced current will be such that it opposes the change in magnetic flux that produced it. This opposition is what causes the 'minus' sign in Faraday's Law.
03

Determine Current Direction for Increasing B

When the magnetic field \( B \) is increasing, the magnetic flux through the loop increases. According to Lenz's Law, the induced current will oppose this increase. Since the external magnetic field is directed into the plane, the induced current will produce a magnetic field out of the plane to oppose the increase, which is achieved by a counterclockwise current in the loop.
04

Determine Current Direction for Decreasing B

When the magnetic field \( B \) is decreasing, the magnetic flux through the loop decreases. Lenz's Law states that the induced current will try to maintain the original magnetic flux by creating additional flux into the plane. This would result in a clockwise induced current.
05

Determine Current Direction for Constant B

If the magnetic field \( B \) is constant with a value \( B_0 \), there is no change in the magnetic flux through the loop. Therefore, no emf is induced and thus no current flows in the loop.

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Key Concepts

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

Lenz's Law
Lenz's Law is a fundamental principle in electromagnetism that explains how induced currents respond to changes in magnetic flux. When a magnetic field through a loop changes, an induced electromotive force (emf) generates a current that opposes the change in magnetic flux. This behavior is a reflection of the conservation of energy and ensures that the resulting electromagnetic actions do not create energy from nowhere. The opposition caused by this law is expressed by the negative sign in Faraday鈥檚 Law equation: \[ \mathcal{E} = -\frac{d\Phi_B}{dt} \].
Here are some key points to remember about Lenz's Law:
  • The direction of the induced current is always such that it opposes the change in magnetic field that causes it.
  • If the magnetic flux increases, the induced current will act in a way that attempts to decrease it.
  • Conversely, if the magnetic flux decreases, the induced current will attempt to increase it.
Induced Current
Induced current is the current generated in a closed circuit due to a changing magnetic field. According to Faraday's Law, any change in the magnetic environment of a coil will induce an emf and thus current in the coil. This principle is crucial in the working of devices like transformers and generators.
Key aspects of induced current include:
  • The magnitude of the induced current is proportional to the rate of change of the magnetic flux.
  • The quicker the change in the magnetic field, the stronger the induced current.
  • If there is no change in magnetic flux, no current will be induced.
The direction of this induced current, as dictated by Lenz's Law, is what determines whether the current flows clockwise or counterclockwise in a loop.
Magnetic Flux
Magnetic flux (\[ \Phi_B \] ) is a measure of the amount of magnetic field passing through a given area, and it plays a pivotal role in electromagnetic induction. The concept can be visualized as the number of magnetic field lines that pass through a closed loop or surface.
To better understand magnetic flux, consider these points:
  • Magnetic flux depends on three key factors: the strength of the magnetic field (\( B \)), the area of the loop (\( A \)), and the angle (\( \theta \)) between the magnetic field and the normal to the surface.
  • Mathematically, it is given by \[ \Phi_B = B \times A \times \cos(\theta) \].
  • A change in any of these parameters can result in a change in magnetic flux and consequently induce an emf in the loop.
Understanding magnetic flux is essential for grasping how and why currents are induced when magnetic fields change.

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Most popular questions from this chapter

CALC A Changing Magnetic Field. You are testing a new data-acquisition system. This system allows you to record a graph of the current in a circuit as a function of time. As part of the test, you are using a circuit made up of a 4.00 -cm-radius, 500 -turn coil of copper wire connected in series to a \(600-\Omega\) resistor. Copper has resistivity \(1.72 \times 10^{-8} \Omega \cdot \mathrm{m},\) and the wire used for the coil has diameter 0.0300 \(\mathrm{mm}\) . You place the coil on a table that is tilted \(30.0^{\circ}\) from the horizontal and that lies between the poles of an electromagnet. The electromagnet generates a vertically upward magnetic field that is zero for \(t<0,\) equal to \((0.120 \mathrm{T}) \times\) \((1-\cos \pi t)\) for \(0 \leq t \leq 1.00 \mathrm{s},\) and equal to 0.240 T for \(t>1.00 \mathrm{s}\) . (a) Draw the graph that should be produced by your data-acquisition system. (This is a full-featured system, so the graph will include labels and numerical values on its axes.) (b) If you were looking vertically downward at the coil, would the current be flowing clockwise or counterclockwise?

CALC In a region of space, a magnetic field points in the \(+x\) -direction (toward the right). Its magnitude varies with position according to the formula \(B_{x}=B_{0}+b x,\) where \(B_{0}\) and \(b\) are positive constants, for \(x \geq 0 .\) A flat coil of area \(A\) moves with uniform speed \(v\) from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

The magnetic field within a long, straight solenoid with a circular cross section and radius \(R\) is increasing at a rate of \(d B / d t\) . (a) What is the rate of change of flux through a circle with radius \(r_{1}\) inside the solenoid, normal to the axis of the solenoid, and with center on the solenoid axis? (b) Find the magnitude of the induced electric field inside the solenoid, at a distance \(r_{1}\) from its axis. Show the direction of this field in a diagram. (c) What is the magnitude of the induced electric field outside the solenoid, at a distance \(r_{2}\).from the axis? (d) Graph the magnitude of the induced electric field as a function of the distance \(r\) from the axis from \(r=0\) to \(r=2 R\) (e) What is the magnitude of the induced emf in a circular turn of radius \(R / 2\) that has its center on the solenoid axis? (f) What is the magnitude of the induced emf if the radius in part (e) is \(R ?\) (g) What is the induced emf if the radius in part (e) is 2R?

Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 165.0 \(\mathrm{cm} /\) its circumference is decreasing at a constant rate of 12.0 \(\mathrm{cm} / \mathrm{s}\) due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 T. (a) Find the emf induced in the loop at the instant when 9.0 s have passed. (b) Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

Back emf. A motor with a brush-and-commutator arrangement, as described in Example \(29.4,\) has a circular coil with radius 2.5 \(\mathrm{cm}\) and 150 turns of wire. The magnetic field has magnitude \(0.060 \mathrm{T},\) and the coil rotates at 440 \(\mathrm{rev} / \mathrm{min.}\) (a) What is the maximum emf induced in the coil? (b) What is the average back emf?
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