The magnetic field B at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T. (The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of -0.0350 T/s. (a) What is the shape of the field lines of the induced electric field shown in Fig. E29.15 , within the colored circle?

Verified step by step guidance

1. Identify the change in the magnetic field: The problem states that the magnetic field is decreasing at a rate of -0.0350 T/s. This indicates that the magnetic field is changing over time.
2. Apply Faraday's Law of Electromagnetic Induction: According to Faraday's Law, a changing magnetic field within a closed loop induces an electromotive force (emf) in the loop. The induced emf generates an electric field.
3. Determine the direction of the induced electric field: Using Lenz's Law, which states that the direction of the induced emf will act to oppose the change in magnetic flux, we can infer the direction of the induced electric field. Since the magnetic field is decreasing, the induced electric field must act in a direction to try to maintain the original magnetic field.
4. Identify the shape of the induced electric field lines: Given that the magnetic field is uniform and directed into the plane, and considering the symmetry of the situation (a circular area), the induced electric field lines will form closed loops that are concentric circles within the colored circle.
5. Understand the orientation of the induced electric field lines: The induced electric field lines will be oriented perpendicular to the direction of the magnetic field (which is into the plane of the diagram). This perpendicular orientation ensures that the electric field lines form concentric circles around the center of the colored circle.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in that loop. The induced EMF is proportional to the rate of change of the magnetic flux through the loop. This principle is fundamental in understanding how electric fields can be generated by varying magnetic fields, which is crucial for analyzing the induced electric field in the given scenario.

Lenz's Law

Lenz's Law complements Faraday's Law by stating that the direction of the induced current (and thus the induced electric field) will be such that it opposes the change in magnetic flux that produced it. This means that if the magnetic field is decreasing, the induced electric field will create a current that attempts to maintain the original magnetic field, leading to a specific orientation of the induced field lines.

Magnetic Field Lines

Magnetic field lines are visual representations of the magnetic field, indicating the direction and strength of the field. The density of these lines represents the strength of the magnetic field, while the direction of the lines shows the field's orientation. In the context of the induced electric field, understanding the shape and direction of these lines helps predict how the induced electric field will behave in response to the changing magnetic field.