A circular loop of wire with radius r = 0.0480 m and resistance R = 0.160 Ω is in a region of spatially uniform magnetic field, as shown in Fig. E29.22. The magnetic field is directed out of the plane of the figure. The magnetic field has an initial value of 8.00 T and is decreasing at a rate of dB/dt = -0.680 T/s. Is the induced current in the loop clockwise or counterclockwise?

검증된 단계별 안내

Identify the direction of the magnetic field: The crosses in the image indicate that the magnetic field is directed into the plane of the figure.

Apply Faraday's Law of Induction: The induced electromotive force (emf) in the loop is given by the rate of change of magnetic flux through the loop. The formula is \( \text{emf} = -\frac{d\Phi_B}{dt} \), where \( \Phi_B = B \cdot A \) is the magnetic flux.

Calculate the change in magnetic flux: Since the magnetic field is decreasing, \( \frac{dB}{dt} = -0.680 \text{ T/s} \). The area \( A \) of the loop is \( \pi r^2 \), where \( r = 0.0480 \text{ m} \).

Determine the direction of the induced current using 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. Since the magnetic field is decreasing, the induced current will create a magnetic field in the same direction as the original field (into the plane).

Conclude the direction of the induced current: To create a magnetic field into the plane, the induced current must flow in a clockwise direction when viewed from above the plane of the loop.

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주요 개념

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

Faraday's Law of Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (EMF) in the wire. The induced EMF is proportional to the rate of change of the magnetic flux, which is the product of the magnetic field strength and the area of the loop. In this scenario, the decreasing magnetic field induces an EMF in the wire loop.

Lenz's Law

Lenz's Law explains the direction of the induced current in a loop. It states that the induced current will flow in a direction that opposes the change in magnetic flux. Since the magnetic field is decreasing, the induced current will create a magnetic field in the same direction as the original field to oppose the decrease, resulting in a counterclockwise current.

Right-Hand Rule

The Right-Hand Rule is a mnemonic for determining the direction of the induced current. When the thumb points in the direction of the magnetic field, the fingers curl in the direction of the induced current. In this case, applying the rule shows that the induced current flows counterclockwise to maintain the magnetic field direction as it decreases.