Ch 29: Electromagnetic Induction

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552당신이 사용하는 게 아니라요?교과서 변경

Young & Freedman Calc 15th Edition

Ch 29: Electromagnetic Induction

문제 34a

29장, 문제 34a

A metal ring 4.50 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. These magnets produce an initial uniform field of 1.12 T between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.250 T/s. What is the magnitude of the electric field induced in the ring?

검증된 단계별 안내

First, understand that the problem involves electromagnetic induction, specifically Faraday's law of induction, which states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in the loop.

Calculate the area of the metal ring. The diameter is given as 4.50 cm, so the radius is half of that. Convert the radius to meters and use the formula for the area of a circle: \( A = \pi r^2 \).

Determine the rate of change of the magnetic flux through the ring. The magnetic flux \( \Phi \) is given by \( \Phi = B \times A \), where \( B \) is the magnetic field and \( A \) is the area. Since the magnetic field is changing at a rate of 0.250 T/s, the rate of change of flux is \( \frac{d\Phi}{dt} = A \times \frac{dB}{dt} \).

Apply Faraday's law of induction, which states that the induced EMF (\( \mathcal{E} \)) is equal to the negative rate of change of magnetic flux: \( \mathcal{E} = -\frac{d\Phi}{dt} \). Substitute the expression for \( \frac{d\Phi}{dt} \) from the previous step.

Finally, relate the induced EMF to the induced electric field \( E \) around the ring. Since the EMF is the work done per unit charge around the loop, and the loop is circular, \( \mathcal{E} = E \times 2\pi r \). Solve for \( E \) using the expression for \( \mathcal{E} \) obtained from Faraday's law.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a circuit induces an electromotive force (EMF) in the circuit. The induced EMF is equal to the negative rate of change of magnetic flux. In this scenario, the changing magnetic field through the metal ring induces an electric field, which can be calculated using this principle.

Magnetic Flux

Magnetic flux refers to the total magnetic field passing through a given area. It is calculated as the product of the magnetic field strength, the area it penetrates, and the cosine of the angle between the field and the normal to the surface. In this problem, the magnetic flux changes as the magnetic field strength decreases, leading to an induced electric field.

Induced Electric Field

An induced electric field arises in a region where the magnetic field changes over time. According to Faraday's Law, this field is related to the rate of change of magnetic flux. The magnitude of the induced electric field in the ring can be determined by considering the rate at which the magnetic field decreases and the geometry of the ring.

관련 실천

교과서 질문

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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교과서 질문

A long, straight solenoid with a cross-sectional area of 8.00 cm² is wound with 90 turns of wire per centimeter, and the windings carry a current of 0.350 A. A second winding of 12 turns encircles the solenoid at its center. The current in the solenoid is turned off such that the magnetic field of the solenoid becomes zero in 0.0400 s. What is the average induced emf in the second winding?

A long, thin solenoid has 400 turns per meter and radius 1.10 cm. The current in the solenoid is increasing at a uniform rate di/dt. The induced electric field at a point near the center of the solenoid and 3.50 cm from its axis is 8.00 × 10^-6 V/m. Calculate di/dt.

A long, thin solenoid has 900 turns per meter and radius 2.50 cm. The current in the solenoid is increasing at a uniform rate of 36.0 A/s. What is the magnitude of the induced electric field at a point near the center of the solenoid and (a) 0.500 cm from the axis of the solenoid; (b) 1.00 cm from the axis of the solenoid?

A cardboard tube is wrapped with two windings of insulated wire wound in opposite directions, as shown in Fig. E29.20. Terminals a and b of winding A may be connected to a battery through a reversing switch. State whether the induced current in the resistor R is from left to right or from right to left in the following circumstances: (a) the current in winding Ais from a to b and is increasing; (b) the current in winding A is from b to a and is decreasing; (c) the current in winding A is from b to a and is increasing.

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교과서 질문

The conducting rod ab shown in Fig. E29.29 makes contact with metal rails ca and db. The apparatus is in a uniform magnetic field of 0.800 T, perpendicular to the plane of the figure. In what direction does the current flow in the rod?

1965

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