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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Verified step by step guidance

1. Understand Faraday's Law of Electromagnetic Induction, which states that a change in the magnetic environment of a coil of wire will induce a voltage in the coil. The law is mathematically represented as $\mathcal{E}=-\frac{d{\mathrm{\Phi }}_B}{dt}$, where $\mathcal{E}$ is the induced electromotive force (EMF) and ${\mathrm{\Phi }}_B$ is the magnetic flux.
2. Consider the assumption that the electric field between the plates of a parallel-plate capacitor drops abruptly to zero at the edges, implying no fringing of the electric field occurs.
3. Imagine a closed loop path as indicated in the problem (dashed line in the figure), which partially lies inside the capacitor (where the electric field is uniform) and partially outside (where the electric field is assumed to be zero).
4. Apply Faraday's law along this loop. Since the loop encloses no time-varying magnetic field, the net EMF around the loop should be zero. Calculate the line integral of the electric field around the loop. Inside the capacitor, the electric field contributes to the integral, but outside, the contribution is zero if we assume no fringing.
5. Analyze the contradiction: if the integral of the electric field over the entire loop is non-zero (due to the contribution inside the capacitor), but Faraday's law predicts zero EMF due to no change in magnetic flux, this contradiction implies that the initial assumption of no fringing is incorrect. Therefore, the electric field must fringe at the edges of the plates to ensure the net EMF around the loop is zero, satisfying Faraday's law.

Key Concepts

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 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 behave in response to changes in charge distribution, particularly at the edges of the plates.

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, especially at the edges, is crucial for analyzing how it can 'fringe' rather than drop abruptly to zero.

Fringing of Electric Fields

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 surrounding space. This occurs because the electric field cannot simply terminate at the edges; instead, it spreads out, leading to a gradual decrease in field strength. Recognizing fringing is essential for understanding the limitations of the idealized model of a capacitor and the implications for real-world applications.