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 16

Chapter 30, Problem 16

A 1000-turn coil of wire 1.0 cm in diameter is in a magnetic field that increases from 0.10 T to 0.30 T in 10 ms. The axis of the coil is parallel to the field. What is the emf of the coil?

Verified step by step guidance

Step 1: Calculate 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 diameter is given as 1.0 cm, so the radius is \( r = \frac{1.0 \text{ cm}}{2} = 0.5 \text{ cm} = 0.005 \text{ m} \). Substitute \( r \) into the formula to find \( A \).

Step 2: Determine the change in magnetic flux \( \Delta \Phi \) through the coil. Magnetic flux is given by \( \Phi = B \cdot A \), where \( B \) is the magnetic field and \( A \) is the area. The change in flux is \( \Delta \Phi = (B_{\text{final}} - B_{\text{initial}}) \cdot A \). Substitute \( B_{\text{initial}} = 0.10 \text{ T} \), \( B_{\text{final}} = 0.30 \text{ T} \), and the area \( A \) calculated in Step 1.

Step 3: Use Faraday's law of electromagnetic induction to calculate the induced emf. Faraday's law states \( \text{emf} = -N \frac{\Delta \Phi}{\Delta t} \), where \( N \) is the number of turns in the coil, \( \Delta \Phi \) is the change in magnetic flux, and \( \Delta t \) is the time interval. Substitute \( N = 1000 \), \( \Delta \Phi \) from Step 2, and \( \Delta t = 10 \text{ ms} = 0.010 \text{ s} \).

Step 4: Simplify the expression for the emf by performing the necessary multiplications and divisions. Ensure the negative sign from Faraday's law is included, which indicates the direction of the induced emf according to Lenz's law.

Step 5: Interpret the result. The magnitude of the emf represents the rate of change of magnetic flux through the coil, and the negative sign indicates the direction of the induced current opposes the change in magnetic flux.

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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 a change in magnetic flux through a coil induces an electromotive force (emf) in the coil. The induced emf is directly proportional to the rate of change of magnetic flux and the number of turns in the coil. Mathematically, it is expressed as emf = -N (ΔΦ/Δt), where N is the number of turns, ΔΦ is the change in magnetic flux, and Δt is the time interval.

Magnetic Flux

Magnetic flux (Φ) is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It is defined as the product of the magnetic field (B) and the area (A) through which the field lines pass, given by Φ = B · A · cos(θ), where θ is the angle between the magnetic field lines and the normal to the surface. In this case, since the coil's axis is parallel to the field, θ is 0 degrees.

Induced EMF Calculation

To calculate the induced emf in the coil, we first determine the change in magnetic flux due to the change in the magnetic field. The area of the coil can be calculated using its diameter, and the change in magnetic field is the difference between the final and initial values. By substituting these values into Faraday's Law, we can find the induced emf, which quantifies the voltage generated in the coil due to the changing magnetic field.

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Related Practice

Textbook Question

FIGURE EX30.13 shows a 10-cm-diameter loop in three different magnetic fields. The loop's resistance is 0.20 Ω. For each, what are the size and direction of the induced current?

Textbook Question

CALC A 5.0-cm-diameter coil has 20 turns and a resistance of 0.50 Ω. A magnetic field perpendicular to the coil is B = 0.020t + 0.010t², where B is in tesla and t is in seconds. Find an expression for the induced current I(t) as a function of time.

Textbook Question

FIGURE EX30.8 shows a 2.0-cm-diameter solenoid passing through the center of a 6.0-cm-diameter loop. The magnetic field inside the solenoid is 0.20 T. What is the magnitude of the magnetic flux through the loop when it is perpendicular to the solenoid and when it is tilted at a 60° angle?

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Textbook Question

A solenoid is wound as shown in FIGURE EX30.9. Is there an induced current as magnet 2 is moved away from the solenoid? If so, what is the current direction through resistor R?

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Textbook Question

FIGURE EX30.19 shows the current as a function of time through a 20-cm-long, 4.0-cm-diameter solenoid with 400 turns. Draw a graph of the induced electric field strength as a function of time at a point 1.0 cm from the axis of the solenoid.

Textbook Question

CALC A 5.0-cm-diameter coil has 20 turns and a resistance of 0.50 Ω. A magnetic field perpendicular to the coil is B = 0.020t + 0.010t², where B is in tesla and t is in seconds. Evaluate I at t = 5 s and t = 10 s.

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