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.
(a) Is the induced current in the loop clockwise or counterclockwise?

Verified step by step guidance

1. Step 1: Identify the direction of the magnetic field. The magnetic field is directed out of the plane of the figure, as indicated by the 'X' symbols.
2. Step 2: Determine the change in the magnetic field. The magnetic field is decreasing at a rate of dB/dt = -0.680 T/s.
3. Step 3: Apply Faraday's Law of Induction. The induced electromotive force (emf) in the loop is given by the negative rate of change of the magnetic flux through the loop: emf = -dΦ/dt.
4. Step 4: Calculate the magnetic flux. The magnetic flux Φ through the loop is given by Φ = B * A, where A is the area of the loop (A = πr^2).
5. Step 5: Determine the direction of the induced current using Lenz's Law. Lenz's Law 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 (out of the plane). Therefore, the induced current will be counterclockwise.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a closed loop induces an electromotive force (EMF) in the loop. The induced EMF is proportional to the rate of change of the magnetic field and the area of the loop. This principle is fundamental in understanding how electric currents can be generated by changing magnetic fields.

Lenz's Law

Lenz's Law provides the direction of the induced current resulting from an induced EMF. It states that the induced current will flow in a direction that opposes the change in magnetic flux that produced it. This law is crucial for determining whether the induced current in the loop is clockwise or counterclockwise in response to the decreasing magnetic field.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength and the area through which the field lines pass, taking into account the angle between the field lines and the normal to the surface. It quantifies the total magnetic field passing through a given area and is essential for applying Faraday's Law to determine the induced EMF in the loop.