The current in Fig. E29.18E29.18 obeys the equation I(t)=I0e−btI(t)=I_$$0e^{-bt}$$, where 0"\u003eb\u003e0b \u003e 0. Find the direction (clockwise or counterclockwise) of the current induced in the round coil for 0"\u003et\u003e0t \u003e 0.

Ch 29: Electromagnetic Induction

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 29: Electromagnetic Induction

Problem 18

Chapter 29, Problem 18

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 $t is greater than 0$ .

Verified step by step guidance

Identify the components in the image: The image shows a round coil labeled 'A' and a straight wire carrying a current 'I' in the downward direction.

Understand the current equation: The current in the wire is given by I(t) = I_0e^(-bt), where I_0 is the initial current and b is a positive constant. This indicates that the current decreases exponentially over time.

Apply Faraday's Law of Induction: A changing current in the straight wire will produce a changing magnetic field, which can induce an electromotive force (emf) in the nearby coil 'A'.

Determine the direction of the induced current: According to Lenz's Law, the induced current in the coil will flow in a direction that opposes the change in magnetic flux. Since the current in the wire is decreasing, the magnetic field through the coil is also decreasing.

Conclude the direction of the induced current: To oppose the decrease in magnetic field, the induced current in the coil will create a magnetic field in the same direction as the original field. Using the right-hand rule, determine that the induced current in the coil will be in the clockwise direction when viewed from above.

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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 coil induces an electromotive force (EMF) in the coil. The induced EMF is proportional to the rate of change of the magnetic flux. In this scenario, as the current decreases over time, the magnetic field associated with it changes, inducing a current in the coil.

Lenz's Law

Lenz's Law explains the direction of the induced current: it will oppose the change in magnetic flux that produced it. If the external current decreases, the induced current will act to maintain the original magnetic field direction, thus determining whether it flows clockwise or counterclockwise in the coil.

Exponential Decay

Exponential decay describes how the current I(t) = I_0e^(-bt) decreases over time, where b > 0. This mathematical model indicates that the current diminishes rapidly initially and slows down as time progresses, affecting the rate of change of the magnetic field and thus the induced current in the coil.

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Related Practice

Textbook Question

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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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 armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 cm. The coil rotates in a magnetic field of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 mV?

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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

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 B₀. Explain your reasoning.

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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?

The current in Fig. E29.18E29.18 obeys the equation I(t)=I0e−btI(t)=I_0e^{-bt}, where b>0b > 0. Find the direction (clockwise or counterclockwise) of the current induced in the round coil for t>0t > 0.

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Key Concepts

Faraday's Law of Induction

Lenz's Law

Exponential Decay

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 $t is greater than 0$ .