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. What is the shape of the field lines of the induced electric field shown in Fig. E29.15 , within the colored circle?
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Verified step by step guidance

1. Understand the concept of electromagnetic induction: When the magnetic field within a region changes, it induces an electric field according to Faraday's law of induction.
2. Recall Faraday's law of induction, which states that a changing magnetic field induces an electromotive force (EMF) in a closed loop. The induced electric field is related to the rate of change of the magnetic field.
3. Consider the symmetry of the problem: The magnetic field is uniform and directed into the plane, and it is decreasing uniformly. This suggests that the induced electric field will have a symmetrical pattern.
4. Apply Lenz's law: The direction of the induced electric field will be such that it opposes the change in the magnetic field. Since the magnetic field is decreasing, the induced electric field will circulate in a direction that would create a magnetic field opposing the decrease.
5. Conclude the shape of the field lines: The induced electric field lines will form concentric circles around the axis of the solenoid, within the colored circle, due to the symmetry and the nature of the changing magnetic field.

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 the loop. The induced EMF is proportional to the rate of change of the magnetic flux through the loop. In this scenario, the decreasing magnetic field inside the circle induces an electric field, which can be visualized as field lines forming closed loops around the changing magnetic field.

Lenz's Law

Lenz's Law provides the direction of the induced electric field and states that the induced EMF will generate a current whose magnetic field opposes the change in the original magnetic field. This means that the induced electric field lines will form closed loops in a direction that counteracts the decrease in the magnetic field within the circle, ensuring conservation of energy.

Magnetic Field Inside a Solenoid

A solenoid is a coil of wire that generates a uniform magnetic field inside it when an electric current passes through. The magnetic field inside a long, thin solenoid is typically uniform and directed along the axis of the solenoid. In this problem, the solenoid's magnetic field is decreasing, which is crucial for understanding the induced electric field's behavior and the shape of its field lines.