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. If the ring is cut at some point and the ends are separated slightly, what will be the emf between the ends?
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Verified step by step guidance

1. Understand that the problem involves electromagnetic induction, specifically Faraday's law of induction, which states that a changing magnetic field within a loop induces an electromotive force (emf) in the loop.
2. Identify that the magnetic field is decreasing at a rate of -0.0350 T/s. This rate of change of the magnetic field is crucial for calculating the induced emf.
3. Use Faraday's law of induction, which is given by the formula: $\epsilon =-\frac{d\Phi /dt}{}$, where $\epsilon$ is the induced emf and $\Phi$ is the magnetic flux.
4. Calculate the magnetic flux $\Phi$ using the formula: $\Phi =BA$, where $B$ is the magnetic field and $A$ is the area of the circle. Since the magnetic field is uniform, the flux is simply the product of the field and the area.
5. Substitute the rate of change of the magnetic field into Faraday's law to find the induced emf: $\epsilon =-A(-0.0350)$. The negative sign in Faraday's law indicates the direction of the induced emf, but since the field is decreasing, the induced emf will be positive.

Key Concepts

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 proportional to the rate of change of the magnetic flux. In this scenario, the decreasing magnetic field within the circle causes a change in magnetic flux, which induces an emf in the ring.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It is calculated as the product of the magnetic field (B) and the area (A) it penetrates, perpendicular to the field. In this problem, the change in magnetic flux due to the decreasing magnetic field is crucial for determining the induced emf.

Lenz's Law

Lenz's Law states that the direction of the induced emf and current will be such that it opposes the change in magnetic flux that produced it. This principle helps determine the polarity of the induced emf in the ring when the magnetic field decreases. It ensures that the induced current creates a magnetic field opposing the reduction in the original field.