Ch 30: Electromagnetic Induction

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 30: Electromagnetic Induction

Problem 38

Chapter 30, Problem 38

A 100-turn, 2.0-cm-diameter coil is at rest with its axis vertical. A uniform magnetic field 60° away from vertical increases from 0.50 T to 1.50 T in 0.60 s. What is the induced emf in the coil?

Verified step by step guidance

Determine the formula for the induced emf using Faraday's Law of Induction: $ \text{emf} = -N \frac{\Delta \Phi_B}{\Delta t} $, where $ N $ is the number of turns, $ \Delta \Phi_B $ is the change in magnetic flux, and $ \Delta t $ is the time interval.

Calculate the magnetic flux $ \Phi_B $ using the formula $ \Phi_B = B A \cos \theta $, where $ B $ is the magnetic field strength, $ A $ is the area of the coil, and $ \theta $ is the angle between the magnetic field and the normal to the coil's surface.

Find the area of the coil using the formula for the area of a circle: $ A = \pi r^2 $, where $ r $ is the radius of the coil. The radius is half the diameter, so $ r = 0.02 \, \text{m} / 2 $.

Calculate the change in magnetic flux $ \Delta \Phi_B $ by substituting the initial and final values of $ B $ into the flux formula: $ \Delta \Phi_B = (B_\text{final} A \cos \theta) - (B_\text{initial} A \cos \theta) $.

Substitute the values for $ N $, $ \Delta \Phi_B $, and $ \Delta t $ into the formula for induced emf to find the result: $ \text{emf} = -N \frac{\Delta \Phi_B}{\Delta t} $. Note that the negative sign indicates the direction of the induced emf according to Lenz's Law.

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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 directly proportional to the rate of change of magnetic flux through the loop. This principle is fundamental in understanding how a changing magnetic field can generate an electric current in a conductor.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength and the area through which the field lines pass, adjusted for the angle between the field and the normal to the surface. It quantifies the total magnetic field passing through a given area and is crucial for calculating the induced emf.

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 that produced it. This law helps determine the polarity of the induced emf and ensures the conservation of energy in electromagnetic systems.

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Lenz's Law

Related Practice

Textbook Question

A 2.0 cm×2.0 cm square loop of wire with resistance 0.010 Ω has one edge parallel to a long straight wire. The near edge of the loop is 1.0 cm from the wire. The current in the wire is increasing at the rate of 100 A/s. What is the current in the loop?

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Textbook Question

At t = 0 s, the current in the circuit in FIGURE EX30.35 is I0. At what time in μs is the current (1/2)I₀?

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Textbook Question

CALC A 10 cm×10 cm square loop of wire lies in the xy-plane. The magnetic field in this region of space is $\vec{B} = (0.30 t \hat{i} + 0.50 t^{2} \hat{k}) T$ , where t is in s. What is the emf induced in the loop at (a) t = 0.5 s and (b) t = 1.0 s?

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Textbook Question

CALC An 8.0 cm×8.0 cm square loop is halfway into a magnetic field perpendicular to the plane of the loop. The loop's mass is 10 g and its resistance is 0.010 Ω. A switch is closed at t = 0 s, causing the magnetic field to increase from 0 to 1.0 T in 0.010 s. Hint: What is the impulse on the loop? With what speed is the loop 'kicked' away from the magnetic field?

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Textbook Question

BIO An MRI machine needs to detect signals that oscillate at very high frequencies. It does so with an LC circuit containing a 15 mH coil. To what value should the capacitance be set to detect a 450 MHz signal?

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Textbook Question

The switch in FIGURE EX30.32 has been in position 1 for a long time. It is changed to position 2 at t = 0 s. What is the first time at which the current is maximum?

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