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 14

Chapter 29, Problem 14

A flat, rectangular coil of dimensions l and w is pulled with uniform speed v through a uniform magnetic field B with the plane of its area perpendicular to the field (Fig. E29.14). (a) Find the emf induced in this coil. (b) If the speed and magnetic field are both tripled, what is the induced emf?

Verified step by step guidance

Understand that the problem involves electromagnetic induction, where a change in magnetic flux through a coil induces an electromotive force (emf). The coil is moving through a magnetic field, which changes the flux.

Recall Faraday's Law of Induction, which states that the induced emf ($ \mathcal{E} $) in a coil is equal to the negative rate of change of magnetic flux ($ \Phi_B $) through the coil: $ \mathcal{E} = -\frac{d\Phi_B}{dt} $.

Calculate the magnetic flux ($ \Phi_B $) through the coil. Since the magnetic field ($ B $) is uniform and perpendicular to the plane of the coil, $ \Phi_B = B \cdot A $, where $ A = l \cdot w $ is the area of the coil.

Determine the rate of change of magnetic flux. As the coil moves with speed $ v $, the area exposed to the magnetic field changes. The rate of change of flux is $ \frac{d\Phi_B}{dt} = B \cdot w \cdot v $, since the length $ l $ of the coil is moving out of the field at speed $ v $.

For part (b), if both the speed $ v $ and the magnetic field $ B $ are tripled, substitute $ 3v $ and $ 3B $ into the expression for the rate of change of flux: $ \frac{d\Phi_B}{dt} = 3B \cdot w \cdot 3v $. Calculate the new induced emf using this updated rate of change.

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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 Electromagnetic Induction

Faraday's Law states that the induced electromotive force (emf) in a closed loop is equal to the negative rate of change of magnetic flux through the loop. For a coil moving through a magnetic field, the emf can be calculated using the formula emf = -dΦ/dt, where Φ is the magnetic flux.

Magnetic Flux

Magnetic flux (Φ) is the measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It is calculated as Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area through which the field lines pass, and θ is the angle between the field lines and the perpendicular to the surface.

Effect of Velocity and Magnetic Field on Induced EMF

The induced emf in a coil moving through a magnetic field is directly proportional to the velocity of the coil and the strength of the magnetic field. If both the speed of the coil and the magnetic field are tripled, the induced emf will increase by a factor of nine, as emf = B * l * v for a rectangular coil moving perpendicular to the field.

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Textbook Question

Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at a constant rate of 12.0 cm/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. Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

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Textbook Question

A closely wound rectangular coil of 80 turns has dimen-sions of 25.0 cm by 40.0 cm. The plane of the coil is rotated from a position where it makes an angle of 37.0° with a magnetic field of 1.70 T to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil?

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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 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 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 $t is greater than 0$ .

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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