Ch. 29 - Electromagnetic Induction and Faraday's Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 29 - Electromagnetic Induction and Faraday's Law

Problem 78

Chapter 28, Problem 78

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.]

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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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Textbook Question

In a certain region of space near Earth’s surface, a uniform horizontal magnetic field of magnitude B exists above a level defined to be y = 0. Below y = 0, the field abruptly becomes zero (Fig. 29–63). A vertical square wire loop has resistivity ρ, mass density ρ_m, diameter d, and side length ℓ. It is initially at rest with its lower horizontal side at y = 0 and is then allowed to fall under gravity, with its plane perpendicular to the direction of the magnetic field. (a) While the loop is still partially immersed in the magnetic field (as it falls into the zero-field region), determine the magnetic “drag” force that acts on it at the moment when its speed is υ. (b) Assume that the loop achieves a constant terminal velocity V_T before its upper horizontal side exits the field. Determine a formula for V_T. (c) If the loop is made of copper and B = 0.80 T, find V_T.

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Textbook Question

What is the energy dissipated as a function of time in a circular loop of 18 turns of wire having a radius of 10.0 cm and a resistance of 2.0 Ω if the plane of the loop is perpendicular to a magnetic field given by B(t) = B₀e⁻ᵗ/ʳ with B₀ = 0.50 T and τ = 0.10 s?

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Textbook Question

A high-intensity desk lamp is rated at 35 W but requires only 12 V. It contains a transformer that converts 120-V household voltage.

(d) What is the resistance of the bulb when on?

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Textbook Question

In an experiment, a coil was mounted on a low-friction cart that moved through the magnetic field B of a permanent magnet. The speed of the cart v and the induced voltage V were simultaneously measured, as the cart moved through the magnetic field, using a computer-interfaced motion sensor and a voltmeter. The Table below shows the collected data:

Make a graph of the induced voltage, V, vs. the speed, v. Determine a best-fit linear equation for the data. Theoretically, the relationship between V and v is given by V = BN𝓁𝓋 where N is the number of turns of the coil, B is the magnetic field, and ℓ is the average of the inside and outside widths of the coil. In the experiment, B = 0.126 T, N = 50, and ℓ = 0.0561 m.

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Textbook Question

Determine the magnetic field at a point P due to a very long wire with a square bend as shown in Fig. 28–63. The point P is halfway between the two corners.

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Textbook Question

Find the % error between the slope of the experimental graph and the theoretical value for the slope.

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