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33. Geometric Optics2h 57m
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35. Special Relativity2h 10m

30. Induction and Inductance

Magnetic Flux

Physics

30. Induction and Inductance

Magnetic Flux

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3:05 minutes

Problem 78

Textbook Question

Apply Faraday’s law, in the form of Eq. 29–8, to show that the static electric field between the plates of a parallel-plate capacitor cannot drop abruptly to zero at the edges, but must, in fact, fringe. Use the path shown dashed in Fig. 29–61. [Hint: Assume the contrary: that there is no fringing. Show that this assumption leads to a contradiction.]

Verified step by step guidance

Faraday's law in integral form is given by: ∮E⋅dl = -dΦB/dt, where E is the electric field, dl is the infinitesimal path element, and ΦB is the magnetic flux. For a static electric field, dΦB/dt = 0, so the equation simplifies to ∮E⋅dl = 0.

Assume, for contradiction, that the electric field between the plates of the capacitor drops abruptly to zero at the edges, with no fringing field. This means the electric field exists only between the plates and is zero outside the region of the plates.

Consider the dashed path shown in Fig. 29–61, which forms a closed loop. Part of the path lies inside the region between the plates (where the electric field is nonzero), and part of the path lies outside the plates (where the electric field is assumed to be zero).

For the path segment inside the plates, the contribution to the line integral ∮E⋅dl is nonzero because the electric field is nonzero. For the path segment outside the plates, the contribution to the line integral is zero because the electric field is assumed to be zero.

Adding these contributions, the total line integral ∮E⋅dl would not equal zero, which contradicts Faraday's law for a static electric field. Therefore, the assumption that the electric field drops abruptly to zero at the edges is incorrect. This implies that the electric field must fringe at the edges of the capacitor to satisfy Faraday's law.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Faraday's Law of Induction

Faraday's Law states that a changing magnetic field within a closed loop induces an electromotive force (EMF) in the wire forming the loop. This principle is fundamental in understanding how electric fields and magnetic fields interact. In the context of capacitors, it helps explain how electric fields can change in response to variations in charge distribution, leading to the concept of fringing.

Electric Field in a Capacitor

The electric field between the plates of a parallel-plate capacitor is uniform and directed from the positive to the negative plate. This field is defined as the force per unit charge experienced by a positive test charge placed in the field. Understanding the behavior of this electric field, particularly at the edges of the plates, is crucial for analyzing how it behaves under different conditions, including the phenomenon of fringing.

Fringing Effects

Fringing refers to the phenomenon where the electric field lines extend beyond the edges of the capacitor plates, creating a non-uniform field in the region surrounding the plates. This occurs because the assumption of a perfectly uniform field breaks down at the edges, leading to a gradual transition rather than an abrupt drop to zero. Recognizing fringing is essential for accurately predicting the behavior of capacitors in practical applications.

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Related Practice

Textbook Question

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 cm² is rotated in 0.040 s from a position where its plane is perpendicular to the earth's magnetic field to a position where its plane is parallel to the field. The earth's magnetic field at the lab location is 6.0 × 10^-5 T. What is the total magnetic flux through the coil before it is rotated? After it is rotated?

Textbook Question

An equilateral triangle 8.0 cm on a side is in a 5.0 mT uniform magnetic field. The magnetic flux through the triangle is 6.0 μWb. What is the angle between the magnetic field and an axis perpendicular to the plane of the triangle?

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Textbook Question

What is the magnitude of the magnetic flux through the loop shown in FIGURE EX30.5?

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Textbook Question

A 16-cm-diameter circular loop of wire is placed in a 0.65-T magnetic field. (a) When the plane of the loop is perpendicular to the field lines, what is the magnetic flux through the loop? (b) The plane of the loop is rotated until it makes a 35° angle with the field lines. What is the angle θ in Eq. 29–1a for this situation? (c) What is the magnetic flux through the loop at this angle?

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Textbook Question

FIGURE EX30.8 shows a 2.0-cm-diameter solenoid passing through the center of a 6.0-cm-diameter loop. The magnetic field inside the solenoid is 0.20 T. What is the magnitude of the magnetic flux through the loop when it is perpendicular to the solenoid and when it is tilted at a 60° angle?

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Multiple Choice

A ring of radius 0.5m lies in the xy-plane. If a magnetic field of magnitude 2 T points at an angle of 22° above the x-axis, what is the magnetic flux through the ring?

A square conducting loop with sides equal to