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 8

Chapter 30, Problem 8

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?

Verified step by step guidance

Step 1: Understand the concept of magnetic flux. Magnetic flux (Φ) is given by the formula Φ = B × A × cos(θ), where B is the magnetic field strength, A is the area through which the field passes, and θ is the angle between the magnetic field and the normal to the surface.

Step 2: Calculate the area of the solenoid cross-section. Since the solenoid has a diameter of 2.0 cm, its radius is 1.0 cm (or 0.01 m). The area of the solenoid cross-section is A_solenoid = π × r² = π × (0.01 m)².

Step 3: Determine the area of the loop. The loop has a diameter of 6.0 cm, so its radius is 3.0 cm (or 0.03 m). The area of the loop is A_loop = π × r² = π × (0.03 m)². However, only the area of the solenoid contributes to the flux through the loop because the solenoid is smaller and passes through the center of the loop.

Step 4: Calculate the magnetic flux when the loop is perpendicular to the solenoid. When the loop is perpendicular, θ = 0°, and cos(0°) = 1. Use the formula Φ = B × A_solenoid × cos(θ) to find the flux.

Step 5: Calculate the magnetic flux when the loop is tilted at a 60° angle. When the loop is tilted, θ = 60°, and cos(60°) = 0.5. Use the same formula Φ = B × A_solenoid × cos(θ), substituting θ = 60° to find the flux in this case.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Magnetic Flux

Magnetic flux 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, and is given by the formula Φ = B · A · cos(θ), where θ is the angle between the magnetic field and the normal to the surface.

Solenoid

A solenoid is a long coil of wire that generates a magnetic field when an electric current passes through it. The magnetic field inside a solenoid is uniform and can be calculated using the formula B = μ₀(nI), where μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current. The strength of the magnetic field is crucial for determining the magnetic flux through nearby loops.

Angle of Inclination

The angle of inclination, or tilt angle, affects the calculation of magnetic flux through a surface. When the surface is perpendicular to the magnetic field, the angle θ is 0°, resulting in maximum flux. As the angle increases, the effective area through which the magnetic field lines pass decreases, which is accounted for in the flux calculation using the cosine of the angle, thus reducing the total magnetic flux.

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Textbook Question

An equilateral triangle 8.0 cm on a side is in a 5.0 mT uniform magnetic field. The magnetic flux through the triangle is 6.0 μWb. What is the angle between the magnetic field and an axis perpendicular to the plane of the triangle?

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

What is the magnitude of the magnetic flux through the loop shown in FIGURE EX30.5?

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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

A potential difference of 0.050 V is developed across the 10-cm-long wire of FIGURE EX30.3 as it moves through a magnetic field perpendicular to the figure. What are the strength and direction (in or out) of the magnetic field?

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Textbook Question

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?