Ch 29: Electromagnetic Induction

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 29: Electromagnetic Induction

Problem 22

Chapter 29, Problem 22

A circular loop of wire with radius r = 0.0480 m and resistance R = 0.160 Ω is in a region of spatially uniform magnetic field, as shown in Fig. E29.22. The magnetic field is directed out of the plane of the figure. The magnetic field has an initial value of 8.00 T and is decreasing at a rate of dB/dt = -0.680 T/s. Is the induced current in the loop clockwise or counterclockwise?

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Identify the direction of the magnetic field: The crosses in the image indicate that the magnetic field is directed into the plane of the figure.

Apply Faraday's Law of Induction: The induced electromotive force (emf) in the loop is given by the rate of change of magnetic flux through the loop. The formula is $ \text{emf} = -\frac{d\Phi_B}{dt} $, where $ \Phi_B = B \cdot A $ is the magnetic flux.

Calculate the change in magnetic flux: Since the magnetic field is decreasing, $ \frac{dB}{dt} = -0.680 \text{ T/s} $. The area $ A $ of the loop is $ \pi r^2 $, where $ r = 0.0480 \text{ m} $.

Determine the direction of the induced current using Lenz's Law: Lenz's Law states that the direction of the induced current will be such that it opposes the change in magnetic flux. Since the magnetic field is decreasing, the induced current will create a magnetic field in the same direction as the original field (into the plane).

Conclude the direction of the induced current: To create a magnetic field into the plane, the induced current must flow in a clockwise direction when viewed from above the plane of the loop.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Faraday's Law of Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (EMF) in the wire. The induced EMF is proportional to the rate of change of the magnetic flux, which is the product of the magnetic field strength and the area of the loop. In this scenario, the decreasing magnetic field induces an EMF in the wire loop.

Lenz's Law

Lenz's Law explains the direction of the induced current in a loop. It states that the induced current will flow in a direction that opposes the change in magnetic flux. Since the magnetic field is decreasing, the induced current will create a magnetic field in the same direction as the original field to oppose the decrease, resulting in a counterclockwise current.

Right-Hand Rule

The Right-Hand Rule is a mnemonic for determining the direction of the induced current. When the thumb points in the direction of the magnetic field, the fingers curl in the direction of the induced current. In this case, applying the rule shows that the induced current flows counterclockwise to maintain the magnetic field direction as it decreases.

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Related Practice

Textbook Question

A metal ring 4.50 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 T/s. What is the magnitude of the electric field induced in the ring?

Textbook Question

Using Lenz's law, determine the direction of the current in resistor ab of Fig. E29.19 when (a) switch S is opened after having been closed for several minutes; (b) coil B is brought closer to coil A with the switch closed; (c) the resistance of R is decreased while the switch remains closed.

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Textbook Question

The magnetic field within a long, straight solenoid with a circular cross section and radius R is increasing at a rate of dB/dt. What is the magnitude of the induced emf in a circular turn of radius R/2 that has its center on the solenoid axis?

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Textbook Question

A cardboard tube is wrapped with two windings of insulated wire wound in opposite directions, as shown in Fig. E29.20. Terminals a and b of winding A may be connected to a battery through a reversing switch. State whether the induced current in the resistor R is from left to right or from right to left in the following circumstances: (a) the current in winding Ais from a to b and is increasing; (b) the current in winding A is from b to a and is decreasing; (c) the current in winding A is from b to a and is increasing.

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Textbook Question

The conducting rod ab shown in Fig. E29.29 makes contact with metal rails ca and db. The apparatus is in a uniform magnetic field of 0.800 T, perpendicular to the plane of the figure. In what direction does the current flow in the rod?

Textbook Question

The current in Fig. $E 29.18$ obeys the equation $I (t) = I_{0} e^{- b t}$ , where $b is greater than 0$ . Find the direction (clockwise or counterclockwise) of the current induced in the round coil for $t is greater than 0$ .

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