The magnetic field B at all points within the colored circle...

The magnetic field $\overrightarrow{B}$ 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? (b) What are the magnitude and direction of this field at any point on the circular conducting ring with radius 0.100 m? (c) What is the current in the ring if its resistance is 4.00 $\Omega$? (d) What is the emf between points $a$ and $b$ on the ring? (e) If the ring is cut at some point and the ends are separated slightly, what will be the emf between the ends?

Key Concepts

Induced Electric Fields (Non-Conservative Fields)

In situations involving time-varying magnetic fields, electric fields are induced that do not arise from static charges and do not obey the usual conservative property. Instead, these non-conservative electric fields create closed loops, often forming circular patterns around regions where the magnetic field is changing. This concept is key to understanding the behavior of induced currents in circuits subject to changing magnetic environments.

Lenz's Law

Lenz's Law provides the direction of the induced emf and current by stating that the induced current will flow in such a direction as to oppose the change in magnetic flux that produced it. This law ensures the conservation of energy and explains why the negative sign appears in Faraday's Law, indicating that the induced effects act to resist the flux change.

Ohm's Law in Electromagnetic Induction

Ohm's Law, which relates current, voltage, and resistance, plays a critical role when analyzing circuits under electromagnetic induction. When an emf is induced in a conductive loop, the resulting current can be calculated using Ohm's Law. This concept is essential for determining quantities like the induced current and the potential difference between points in the circuit, especially when there is resistance present.

Magnetic Flux

Magnetic flux is the measure of the total magnetic field passing through a given area. It plays a central role in electromagnetic induction, where any change in the magnetic flux through a closed loop induces an electromotive force (emf). This concept is fundamental to understanding how energy is transferred from changing magnetic fields to electric circuits.

Faraday's Law of Induction

Faraday's Law quantitatively describes how a time-varying magnetic flux through a closed loop induces an emf in that loop. It states that the induced emf is equal to the negative rate of change of the magnetic flux. This principle is the cornerstone of electromagnetic induction and is used in systems ranging from electrical generators to transformers.

Transcript

00:01 Part a of the given problem, the induced electric field is in the direction of induced current, the direction of the magnetic field is into the page and decreasing, so the magnetic flux is also decreasing.

00:18 According to the lens law, the direction of the induced magnetic field must be the same direction of the original magnetic field, which is into the page.

00:27 So according to right handle, the direction of the induced current is clockwise.

00:35 Hence, the direction of induced electric field is a clockwise.

00:42 The electric field is tangent to the ring and its direction is clockwise.

00:47 As we can see from the figure here, let me draw the figure.

00:51 So here we have magnetic field and magnetic field here.

01:00 So from a circle here, so let's say this is the bigger circle with radius r here and the electric field is here inside e with making a radius of a small r here.

01:24 Then for the part a, so the electric field here in the for part a, sorry, is tangent to the ring and its direction is clockwise.

01:38 So here is electric field clockwise.

01:41 Part b, now we need to find the magnitude of the induced electrophil at the r is equal to capital r.

01:48 The induced electric fuel is given by ferrarian's law that says in the ucmph is integral of e .d .l.

01:58 That is a negative rate of a flux change d t.

02:02 The magnetic flux through circle radius r is flux is equal to area times v which is pi r square times b.

02:12 The integration on the left hand side gives us integral of e .d .l is equal to electric field times the part which is a 2 pi r circumference.

02:27 Then we can write electric field e2 pi r is equal to pi r square times the db divided by d t.

02:37 From here, electric field will be r divided by 2, db divided by d t...

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