Physics Solution Manual for 1100 and 2101

Solving for the number of turns n per unit length

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Unformatted text preview: is, then, µ n ∆I ∆B NA 0 ∆t ∆t = R R NA I= (4π × 10−7 T ⋅ m/A)(400 turns/m)(0.40 A) (0.050 s) = = 1.6 × 10−5 A 1.5 Ω ______________________________________________________________________________ (10)(6.0 × 10−4 m 2 ) 32. REASONING The external magnetic field is perpendicular to the plane of the horizontal loop, so it must point either upward or downward. We will use Lenz’s law to decide whether the external magnetic B field points up or down. This law predicts that the direction of the induced magnetic field Bind opposes the change in the magnetic flux through the loop due to the external field. SOLUTION The external magnetic field B is increasing in magnitude, so that the magnetic flux through the loop also increases with time. In order to oppose the increase in magnetic flux, the induced magnetic field Bind must be directed opposite to the external magnetic field B. The drawing shows the loop as viewed from above, with an induced current Iind flowing clockwise. According to Right-Hand Rule No. 2 (see Section 21.7), this induced current creates an induced magnetic field Bind that is Iind Bind (into page) B (out of page) Chapter 22 Problems 1213 directed into the page at the center of the loop (and all other points of the loop’s interior). Therefore, the external magnetic field B must be directed out of the page. Because we are viewing the loop from above, “out of the page” corresponds to upward toward the viewer. 33. SSM REASONING AND SOLUTION If the north and south poles of the magnet are interchanged, the currents in the ammeter would simply be reversed in direction. This follows from Lenz’s law. As the south pole approaches the coil in Figure 22.1b, the field that the coil experiences is becoming stronger and points toward the south pole. To oppose the increasing flux from this field (as specified by Lenz’s law) the current induced in the coil must produce a field that points away from the approaching south pole. RHR-2 indicates that to do this, the induced current must flow into the ammeter on the right and out of the ammeter on the left. Another way to look at things is to realize that, with this direction for the induced current, the coil becomes an electromagnet with a south pole located at its left side. This induced south pole opposes the motion of the approaching south pole of the moving magnet. Therefore, in Figure 22.1b, the current will flow right to left through the ammeter. Similar reasoning (with the poles of the magnet interchanged in Figure 22.1) leads to the conclusion that in Figure 22.1c the induced current must flow into the ammeter on the left and out of the ammeter on the right. Thus, in Figure 22.1c, the current will flow left to right through the ammeter. ______________________________________________________________________________ 34. REASONING The magnetic field produced by I extends throughout the space surrounding the loop. Using RHR-2, it can be shown that the magnetic field is parallel to the normal to the loop. Thus, the magnetic field penetrates the loop and generates a magnetic flux. According to Faraday’s law of electromagnetic induction, an emf is induced when the magnetic flux through the loop is changing in time. If the current I is constant, the magnetic flux is constant, and no emf is induced in the loop. However, if the current is decreasing in time, the magnetic flux is decreasing and an induced current exists in the loop. Lenz’s law states that the induced magnetic field opposes the change in the magnetic field produced by the current I. The induced magnetic field does not necessarily oppose the magnetic field itself. Thus, the induced magnetic field does not always have a direction that is opposite to the direction of the field produced by I. SOLUTION At the location of the loop, the magnetic field produced by the current I is directed into the page (this can be verified by using RHR-2). The current is decreasing, so the magnetic field is decreasing. Therefore, the magnetic flux that penetrates the loop is decreasing. According to Lenz’s law, the induced emf has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes this flux 1214 ELECTROMAGNETIC INDUCTION change. The induced magnetic field will oppose this decrease in flux by pointing into the page, in the same direction as the field produced by I. According to RHR-2, the induced current must flow clockwise around the loop in order to produce such an induced field. The current then flows from left-to-right through the resistor. ______________________________________________________________________________ 35. SSM REASONING In solving this problem, we apply Lenz's law, which essentially says that the change in magnetic flux must be opposed by the induced magnetic field. SOLUTION a. The magnetic field due to the wire in the vicinity of position 1 is directed out of the paper. The coil is moving closer to the wire into a region of higher magnetic field, so the flux through the coil is increasing. Lenz’s law demands that the induced field counteract this increase. The direction of th...
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